Characterising $k$-connected sets in infinite graphs

A $k$-connected set in an infinite graph, where $k>0$ is an integer, is a set of vertices such that any two of its subsets of the same size $\ell \leq k$ can be connected by $\ell$ disjoint paths in the whole graph. We characterise the existence of $k$-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. This characterisation provides an analogue of the Star-Comb Lemma, one of the most-used tools in topological infinite graph theory, for higher connectivity. We also prove a duality theorem for the existence of such $k$-connected sets: if a graph contains no such a $k$-connected set, then it has a tree structure which, whenever it exists, clearly precludes the existence of such a $k$-connected set.


§1. Introduction
It is a well-known and easy-to-prove fact that every connected finite graph contains a long path or a vertex of high degree.More precisely, for every m P N there is an n P N such that each connected graph with at least n vertices either contains a path P m of length m or a star K 1,m with m leaves as a subgraph (cf.[3,Prop. 9.4.1]).In a sense which can be made precise [3,Thm. 9.4.5], the existence of these 'unavoidable' subgraphs characterises connectedness with respect to the subgraph relation in a minimal way: in every infinite collection consisting of graphs with arbitrary large connected subgraphs we find arbitrary long paths or arbitrary large stars as subgraphs; but with only paths or only stars this would not be true, since long paths and large stars do not contain each other.
For 2-connected graphs there is an analogous result, which also is folklore: For every m P N there is an n P N such that every 2-connected finite graph with at least n vertices either contains a subdivision of a cycle C m of length m or a subdivision of a complete bipartite graph K 2,m [3,Prop. 9.4.2].As before, cycles and K 2,m 's as unavoidable topological minors characterise 2-connectedness with respect to the topological minor relation in a minimal way: in every infinite collection consisting of graphs with arbitrary large subdivisions of 2-connected graphs we find arbitrary long cycles or arbitrary large K 2,m 's as topological minors, with the minimality as above [3,Thm. 9.4.5].
In 1993, Oporowski, Oxley and Thomas [15] continued on this path and studied unavoidable subdivisions for different notions of 'k-connectedness' for k P t3, 4u.They gave a finite list of 3-connected graphs that are unavoidable subdivisions in 3-connected graphs, as well as a finite list of internally 4-connected † graphs that are unavoidable subdivisions in internally 4-connected graphs.In [3, Thm.9.4.3 and Thm.9.4.4],Diestel noted that, if one considers minors instead of subdivisions, these two lists can be simplified to be lists of k-connected graphs that characterise the usual notion of k-connectivity with respect to the minor relation, again in a minimal way as above [3,Thm. 9.4.5].For k " 3 the unavoidable minors are the wheel or the complete bipartite graph K 3,m , while for k " 4 the number of unavoidable minors is growing to four different minors, whose definition we omit here.
Recently, Geelen and Joeris [8,13] generalised these results to arbitrary k P N. For this, they again relaxed 'k-connectedness' to another well-known notion.
For k P N, a set X of at least k vertices of a graph G is called k-connected in G, if for all Z 1 , Z 2 Ď X with |Z 1 | " |Z 2 | ď k there are |Z 1 | many vertex disjoint paths from Z 1 to Z 2 in G.Note that any subset Y Ď X with |Y | ě k is also k-connected in G.We often omit stating the graph in which X is k-connected if it is clear from the context.It can be easily seen that any set containing precisely one vertex of each branch set of a k-connected set in a minor of G is k-connected in G (cf. Lemma 4.2).Moreover, the non-existence of a large k-connected set in a graph is a property which is closed under both the minor and topological minor relation.Hence the existence of k-connected sets in a graph is a property which is well-suited to be characterised via the existence of certain (topological) minors.
Geelen and Joeris [8,13] introduced certain graphs called generalised wheels (depending on k and m), which together with the complete bipartite graph K k,m are the unavoidable minors: they contain large k-connected sets themselves and they characterise graphs that contain large k-connected sets with respect to the minor relation.
Since the vertex set of a k-connected graph is a k-connected set in that graph, the characterisation results for k-connectivity as above for k P t1, 2, 3, 4u can also be viewed as characterisations for graphs containing large k-connected sets.And for k P t2, 3, 4u the generalised wheels together with the complete bipartite graph K k,m correspond precisely to the unavoidable minors mentioned before.Now let us consider infinite graphs.Again there is a well-known and easy-to-prove fact that each infinite connected graph contains either a ray, that is a one-way infinite path, or a vertex of infinite degree.This can also be seen as a characterisation of infinite connected graphs via these two unavoidable subgraphs: a ray and the complete bipartite graph K 1,ℵ 0 .
There is also a more localised version of this result, which is known as the Star-Comb † Although we will not use this definition later, a graph is called internally 4-connected if it is 3-connected and, for every separation pA, Bq of order 3, one of A B or B A contains at most one vertex.
Lemma (cf.Lemma 2.5).In essence this lemma relates these subgraphs to a given vertex set.
For 2-connected infinite graphs one can easily construct an analogous result.A double ray is a two-way infinite path.We say a vertex d dominates a ray R if they cannot be separated by deleting a finite set of vertices not containing d.An end of a graph is an equivalence class of rays, where two rays are equivalent, if they cannot be separated by deleting a finite set of vertices.Note that between two rays in the same end there are infinitely many pairwise disjoint paths between those rays by an infinite version of Menger's Theorem (cf.Theorem 2.2).Given a ray R and the complete graph on two vertices K 2 , the one-way infinite ladder is the graph R ˆK2 ‡ .One can easily construct a subdivision of the one-way infinite ladder starting with two disjoint rays in the same end by successively adding paths disjoint to suitable initial segments of the rays as well as the previously chosen paths.Now it is a common exercise to prove that every infinite 2-connected graph contains either a double ray whose subrays belong to the same end, a ray which is dominated by a vertex, or a subdivision of a K 2,ℵ 0 .Hence we obtain a characterisation of 2-connectivity by three unavoidable topological minors: the one-way infinite ladder, the union of a ray R with a complete bipartite graph between a single vertex and V pRq as well as the complete bipartite graph K 2,ℵ 0 .With the advent of topological infinite graph theory, those results became an even more meaningful extension of the finite result.In locally finite graphs, that are graphs where each vertex has finite degree, a double ray whose subrays belong to the same end is the easiest example of an infinite topological circle, that is a homeomorphic image of the sphere S 1 in the Freudenthal compactification of the 1-complex of G (cf. [3,Section 8.6]).
Moreover, a similar topological approach works in finitely separable graphs, that are graphs containing no subdivision of K 2,ℵ 0 .In such a graph, a ray starting at a vertex dominating it is also an infinite topological circle [4, Section 5].
In 1978, Halin [11] studied such a problem for arbitrary k P N.He showed that every k-connected graph whose set of vertices has size at least κ for some uncountable regular cardinal κ contains a subdivision of K k,κ .Hence for all those cardinals, K k,κ is the unique unavoidable subdivision characterising graphs with k-connected subdivisions of size κ.In a way, this characterisation result is stronger than the results previously discussed, since it obtains a direct equality between the size k-connected set and the size of the minors which was not possible for finite graphs.The unavoidable (topological) minors for graphs whose set of vertices has singular cardinality remained undiscovered.
Oporowski, Oxley and Thomas [15] also studied countably infinite graphs for arbitrary k P N, but again for a different notion of 'k-connectedness'.Together with the K k,ℵ 0 , ‡ A definition of the Cartesian product of graphs is given in Section 2.
the unavoidable minors for countably infinite essentially k-connected § graphs have the following structure.For , d P N with `d " k, they consist of a set of disjoint rays, d vertices that dominate one of the rays (or equivalently all of those rays) and infinitely many edges connecting pairs of them in a tree-like way.
This leads to the first part of our main result.For k P N and an infinite cardinal κ we will define certain graphs with a k-connected set of size κ in Section 3, the so called k-typical graphs.These graphs will encompass complete bipartite graphs K k,κ as well as the graphs described by Oporowski, Oxley and Thomas [15] for κ " ℵ 0 .We will moreover introduce such graphs even for singular cardinals κ.It will turn out that for fixed k and κ there are only finitely many k-typical graphs up to isomorphisms.We shall characterise graphs with a k-connected set of size κ via the existence of a minor of such a k-typical graph with a k-connected set of size κ.In contrast to the finite case, the minimality of the list of these graphs in the characterisation is implied by the fact that it really is a finite list of graphs (for which even if it were not minimal we could pick a minimal sublist), and not a finite list of 'classes of graphs', like 'paths' and 'stars' in the finite case for k " 1.
Moreover we will extend the definition of k-typical graphs to so called generalised ktypical graphs.As before for fixed k and κ there are only finitely many generalised k-typical graphs up to isomorphisms, and we shall extend the characterisation from before to be with respect to the topological minor relation.
In finite graphs, k-connected sets have also been studied in connection to tree-width.
Diestel, Gorbunov, Jensen and Thomassen [6,Prop. 3] showed that for any graph G and k P N, if G contains a pk `1q-connected set of size at least 3k, then G has tree-width at least k, and conversely if G has no pk `1q-connected set ¶ of size at least 3k, then G has tree-width less than 4k.
In infinite graphs, different notions of decompositions of graphs in a tree-like way that extend the notion of tree-decompositions in finite graphs have been studied.Robertson, Seymour and Thomas [16] gave a survey of different results characterising the existence of different kinds of these decompositions via forbidden minors.In recent years, one of those decomposition notions, the notion of a nested set of separations has been studied in more detail [5,10].They correspond to tree-decompositions of finite graphs in a natural § A graph is essentially k-connected if there is a constant c P N such that for each separation pA, Bq of order less than k one of A or B has size less than c.As before, we will not use this notion in this paper.¶ In fact, the authors show something slightly stronger, requiring for the second part the slightly weaker property of a vertex set A in G to be externally k-connected in G, i.e. k-connected in G with the additional property that the required paths can be chosen such that no inner vertex meets A and they do not use edges of GrAs.
way and offer a generalisation for infinite graphs.We define separations and the necessary terms, including the notion of parts for a nested set of separations, which provides some analogue of tree-width, in Section 2. These nested separation systems shall allow us to prove a duality theorem.More precisely, if a graph G does not contain a k-connected set of size κ (and hence contains no k-typical graph as a minor), then G can be decomposed by a nested set of separations of order less than k where every part has size less than κ.The existence of such a decomposition is a trivial obstruction for the existence of a k-connected set of size κ.
This leads to our main theorem.
Theorem 1.Let G be an infinite graph, let k P N and let κ ď |V pGq| be an infinite cardinal.
Then the following are equivalent.
(a) V pGq contains a subset of size κ that is k-connected in G.
(b) G contains a k-typical graph of size κ as a minor with finite branch sets.
(c) G contains a subdivision of a generalised k-typical graph of size κ.
(d) There is no nested set of separations of order less than k of G such that every part has size less than κ.
In fact, we will prove a slightly stronger result which will require some more notation, Theorem 2 in Subsection 3.3.In the same vein as the Star-Comb Lemma, that result will relate the minors (or subdivisions) with a specific k-connected set in the graph.
After fixing some notation and recalling some basic definitions and simple facts in Section 2, we will define the k-typical graphs and generalised k-typical graphs in Section 3.
In Section 4 we will collect some basic facts about k-connected sets and their behaviour with respect to minors or topological minors.Section 5 deals with the structure of ends in graphs.
Subsection 5.1 is dedicated to extend a well-known connection between minimal separators and the degree of an end from locally finite graphs to arbitrary graphs.Afterwards, Subsection 5.2 gives a construction of how to find disjoint rays in some end with additional structure between them.Sections 6 and 7 are dedicated to prove the characterisation via minors and topological minors.The case of κ being a regular cardinal is covered in Section 6 and, respectively, the case of κ being a singular cardinals is covered in Section 7.
In Section 8 we will talk about some applications of the minor characterisation, and in Section 9 we shall prove the duality theorem.§2.Preliminaries In this paper, we will work in ZFC.For general notation about graph theory that we do not specifically introduce here we refer the reader to [3].
In this paper we consider both finite and infinite cardinals.As usual, for an infinite cardinal κ we define its cofinality, denoted by cf κ, as the smallest infinite cardinal λ such that there is a set X Ď tY Ď κ | |Y | ă κu such that |X| " λ and Ť X " κ.We distinguish infinite cardinals κ to regular cardinals, i.e. cardinals where cf κ " κ, and singular cardinals, i.e. cardinals where cf κ ă κ.Note that cf κ is always a regular cardinal.For more information on infinite cardinals and ordinals, we refer the reader to [14].Unless otherwise specified, a path in this paper is a finite graph.The length of a path is the size of its edge set.A path is trivial, if it only contains only one vertex, which we will call its endvertex.Otherwise, the two vertices of degree 1 in the path are its endvertices.
The other vertices are called the inner vertices of the path.
Let A, B Ď V pGq be two (not necessarily disjoint) vertex sets.An A -B path is a path whose inner vertices are disjoint from A Y B such that one of its end vertices lies in A and the other lies in B. In particular, a trivial path whose endvertex is in A X B is also an For an end ω P ΩpGq, let ∆pωq denote degpωq `dompωq, which we call the combined degree of ω.Note that the sum of an infinite cardinal with some other cardinal is just the maximum of the two cardinals.
The following lemma due to König about the existence of a ray is a weak version of the compactness principle in combinatorics.
Lemma 2.3 (König's Infinity Lemma).[3, Lemma 8.1.2]Let pV i q iPN be a sequence of disjoint non-empty finite sets, and let G be a graph on their union.Assume that for every n ą 0 each vertex in V n has a neighbour in V n´1 .Then G contains a ray v 0 v 1 . . .with v n P V n for all n P N.
In Section 9 we shall also use a stronger version of the compactness principle in combinatorics, the Generalised Infinity Lemma.In order to state that lemma, we need the following definitions.
A partially ordered set pP, ďq is directed if any two elements have a common upper bound, i.e. for any p, q P P there is an r P P with p ď r and q ď r.A directed inverse system consists of a directed poset P , a family of sets pX p : p P P q, and for all p, q P P with p ă q a map f q,p : X q Ñ X p such that the maps are compatible, i.e. f q,p ˝fr,q " f r,p for all p, q, r P P with p ă q ă r.The inverse limit of such a directed inverse system is the set lim ÐÝ pX p : p P P q " # px p : p P P q P ź pPP X p : f q,p px q q " x p + .
Lemma 2.4 (Generalised Infinity Lemma).[3, Appendix A] The inverse limit of any directed inverse system of non-empty finite sets is non-empty.
A comb C is the union of a ray R together with infinitely many disjoint finite paths each of which has precisely one vertex in common with R, which has to be an endvertex of that path.The ray R is the spine of C and the end vertices of the finite paths that are not on R together with the end vertices of the trivial paths are the teeth of C. A comb whose spine is in ω is also called an ω-comb.A star is the complete bipartite graph K 1,κ for some cardinal κ, where the vertices of degree 1 are its leaves and the vertex of degree κ is its centre.
Next we state a version of the Star-Comb lemma in a slightly stronger way than elsewhere in the literature (e.g.[3,Lemma 8.2.2]).We also give a proof for the sake of completeness.(a) There is a subset If (a) holds, then we take a tree T Ď G containing U 1 such that each edge of T lies on a path between two vertices of U 1 .Such a tree exists by Zorn's Lemma since U 1 is 1-connected in G.We distinguish two cases.
If T has a vertex c of degree κ, then this yields a subdivided star with centre c and a set U 2 Ď U 1 of leaves with |U 2 | " κ by extending each incident edge of c to a c -U 1 path.
Hence we assume T does not contain a vertex of degree κ.Given some vertex v 0 P V and n P N, let D n denote the vertices of T of distance n to v 0 .Since T is connected, the union Ť tD n | n P Nu equals V pT q.And while κ is regular, it follows that κ " ℵ 0 , and therefore that T is locally finite.Hence each D n is finite and, since T is still infinite, each D n is non-empty.Thus T contains a ray R by Lemma 2.3.If R does not already contain infinitely many vertices of U 1 , then by the property of T there are infinitely many edges of T between V pRq and V pT ´Rq.We can extend infinitely many of these edges to a set of disjoint R -U 1 paths, ending in an infinite subset U 2 Ď U 1 , yielding the desired comb.
In both cases, U 2 is still 1-connected, and hence serves as a candidate for U 1 as well, yielding the "moreover" part of the claim.
The following immediate remark helps to identify when we can obtain stars by an application of the Star-Comb lemma.
Remark 2.6.If there is an ω-comb with teeth U and if v dominates ω, then there is also that G contains a subdivided star with leaves U 1 and centre v.
We say that an end ω is in the closure of a set U Ď V pGq, if there is an ω-comb whose teeth are in U .Note that this combinatorial definition of closure coincides with the topological closure when considering the topological setting of locally finite graphs mentioned in the introduction [3,Section 8.6;4].
For an end ω of G and an induced subgraph G 1 of G we write ωaeG 1 for the set of rays R P ω which are also rays of G 1 .The following remarks are immediate.
(1) ωaeG 1 is an end of G 1 for every end ω P ΩpGq.
(3) The degree of ω P ωpGq in G is equal to the degree of ωaeG 1 in G 1 .
Given an end ω P ΩpGq, we say that an ω-ray R is ω-devouring if no ω-ray is disjoint from R. We need the following lemma about the existence of a single ω-devouring ray for an end ω of at most countable degree, which is a special case of [9, Thm.1].Lemma 2.8.If degpωq ď ℵ 0 for ω P ΩpGq, then G contains an ω-devouring ray.
A different way to prove this lemma arises from the construction of normal spanning trees, cf.[3,Prop. 8.2.4].Imitating this proof according to an enumeration of the vertices of a maximal set of disjoint ω-rays yields that the normal ray constructed this way is ω-devouring.
Let us fix some notations regarding minors.Let G and M be graphs.We say M is a minor of G if G contains an inflated subgraph H Ď G witnessing this, i.e. for each v P V pM q ‚ there is a non-empty branch set Bpvq Ď V pHq; ‚ HrBpvqs is connected; ‚ there is an edge between v, w P V pM q in M if and only if there is an edge between some vertex in Bpvq and a vertex in Bpwq in H.
We call M a finite-branch-set minor or fbs-minor of G if each branch set is finite.
Without loss of generality we may assume that such an inflated subgraph H witnessing that M is a minor of G is minimal with respect to the subgraph relation.Then H has the following properties for all v, w P V pM q: ‚ HrBpvqs is a finite tree T v ; ‚ for each v, w P V pM q there is a unique edge e vw in EpHq between Bpvq and Bpwq if vw P EpM q, and no such edge if vw R EpM q; ‚ each leaf of T v is an endvertex of such an edge between two branch sets.
Given a subset C Ď V pM q and a subset A Ď V pGq, we say that M is an fbs-minor of G with A along C, if M is an fbs-minor of G such that the map mapping each vertex of the inflated subgraph to the branch set it is contained in induces a bijection between A and the branch sets of C. As before, we assume without loss of generality that an inflated subgraph H witnessing that M is an fbs-minor of G is minimal with respect to the subgraph relation.We obtain the properties as above, but a leaf of T v could be the unique vertex of A in Bpvq instead.
For , k P N, we write r , ks for the closed integer interval ti P N | ď i ď ku as well as rk, q for the half open integer interval ti P N | ď i ă ku.Given some set I, a family F indexed by I is a sequence of the form pF i | i P Iq, where the members F i are some not necessarily different sets.For convenience we sometimes use a family and the set of its members with a slight abuse of notation interchangeably, for example with common set operations like Ť F.
Given some J Ď I, we denote by FaeJ the subfamily 1 for all i P I.For a family pF i | i P Nq with index set N we say some property holds for eventually all members, if there is some N P N such that the property holds for F i for all i P N with i ě N .
The following lemma is a special case of the famous Delta-Systems Lemma, a common tool of infinite combinatorics.
Lemma 2.9.[14, Thm.II.1.6]Let κ be a regular cardinal, U be a set and It is easy to check that the union of all parts cover the vertex set of G.Moreover, we allow the empty set ∅ as a nested separation system.In this case, we say that V pGq is a part of ∅ (this can be viewed as the empty intersection of vertex sets of the empty set as an orientation of ∅).
A nested separation system N has adhesion less than k if all separations it contains have order less than k, i.e.N Ď S k pGq.
Note that each oriented edge of the tree of a tree-decomposition of G induces a separation pA, Bq where A is the union of the parts on one side of the edge while B is the union of the parts on the other side of the edge.It is easy to check that the set of separations induced by all those edges is a nested separation system.Moreover, properties like adhesion and the size of parts are transferred by this process.
For more information on nested separation systems and their connection to treedecompositions we refer the interested reader to [5,10].
In Section 9 we will make use of the existence of k-lean tree-decompositions for finite graphs to prove our desired duality theorem, which are closely related to k-connected sets.
Given k P N, a tree-decomposition of adhesion less than k is called k-lean if for any two (not necessarily distinct) parts V t 1 , V t 2 of the tree-decomposition and vertex sets in G or there is an edge tt 1 on the t 1t 2 path in the tree inducing a separation of order less than .In particular, given a k-lean tree-decomposition, each part V t is mintk, |V t |u-connected in G.
In [2], the authors noted that the proof given in [1] of a theorem of Thomas [18,Thm. 5] about the existence of lean tree-decompositions witnessing the tree-width of a finite graph can be adapted to prove the existence of a k-lean tree-decomposition of that graph.This definition can easily be lifted to nested separation systems.A nested separation system N Ď S k pGq is called k-lean if given any two (not necessarily distinct) parts P 1 , P 2 of N and vertex sets in G or there is a separation pA, Bq in N with P 1 Ď A and P 2 Ď B of order less than .Here, we specifically allow the empty set as a nested separation system to be k-lean if its part, the whole vertex set of G, is mintk, |V pGq|u-connected.Again, we obtain that each part P of a k-lean nested separation system is mintk, |P |u-connected in G.
Moreover, note that the nested separation system that a k-lean tree-decomposition induces is k-lean as well.§3.Typical graphs with k-connected sets Throughout this section, let k P N be fixed.Let κ denote an infinite cardinal.
In Subsection 3.1 we shall describe an up to isomorphism finite class of graphs each of which contains a designated k-connected set of size κ.We call such a graph a k-typical graph and the designated k-connected set its core.These graphs will appear as the minors Theorem 1(b).
In Subsection 3.2 we shall describe based on these k-typical graphs a more general but still finite class of graphs each of which again contains a designated k-connected set of size κ.We call such a graph a generalised k-typical graph and the designated k-connected set its core.These graphs will appear as the topological minors Theorem 1(c).

k-typical graphs.
The most basic graph with a k-connected set of size κ is a complete bipartite graph K k,κ " Kpr0, kq, Zq for any infinite cardinal κ and a set Z of size κ disjoint from r0, kq.
Although in this graph the whole vertex set is k-connected, we only want to consider the infinite side Z as the core CpK k,κ q of K k,κ , cf. Figure 3.1.This is the first instance of a k-typical graph with a core of size κ.For uncountable regular cardinals κ, this is the only possibility for a k-typical graph with a core of size κ.For any regular k-blueprint B " pB, D, cq the graph T k pBq is a k-typical graph with a countable core.Such graphs are besides the complete bipartite graph K k,ℵ 0 the only other k-typical graphs with a core of size ℵ 0 .
Note that given two regular k-blueprints B 1 " pB 1 , D 1 , c 1 q and B 2 " pB 2 , D 2 , c 2 q such that there is an isomorphism ϕ between B 1 and B 2 that maps D 1 to D 2 , then T k pB 1 q and T k pB 2 q are isomorphic.Moreover, if ϕ maps c 1 to c 2 , then there is an isomorphism between T k pB 1 q and T k pB 2 q that maps the core of T k pB 1 q to the core of T k pB 2 q.Hence up two isomorphism there are only finitely many k-typical graphs with a core of size ℵ 0 .
Note that given any I Ď cf κ with |I| " cf κ there is a unique order-preserving bijection between cf κ and I. Hence we can relabel subsequence KaeI of a good κ-sequence K to a good κ-sequence KaeI.Moreover, note that any cofinal sequence can be made into a good κ-sequence by looking at an strictly ascending subsequence starting above the cofinality of κ, then replacing each element in the sequence by its successor cardinal and relabel as above.Here we use the fact that each successor cardinal is regular.Hence for every singular cardinal κ there is a good κ-sequence.
Let K " pκ α ă κ | α P cf κq be a good κ-sequence and let ď k be a non-negative integer.
As a generalisation of the graph K k,κ we first consider the disjoint union of the complete bipartite graphs K k,κα .Then we identify sets of vertices each consisting of a vertex of the finite side of each graph, and connect the other k ´ vertices of each with disjoint stars K 1,cf κ .
More formally, let X " r , kq ˆt0u, and for each α P cf κ let Y α " tαu ˆr0, kq ˆt1u and let Z α " tαu ˆκα ˆt2u.We denote the family pY α | α P cf κq with Y and the family pZ α | α P cf κq with Z. Then consider the union Ť tKpY α , Z α q | α P cf κu of the complete bipartite graphs and let -Kpk, Kq denote the graph where for each i P r0, q we identify the set cf κ ˆtiu ˆt1u to one vertex in that union.For this graph we fix some further notation.Let ‚ x i denote pi, 0q P X for i P r , kq; ‚ y i " y α i for all α P cf κ denote the vertex corresponding to cf κ ˆtiu ˆt1u for i P r0, q; we call such a vertex a degenerate vertex of -Kpk, Kq; ‚ y α i denote pα, i, 1q for i P r , kq; and ‚ Y i denote py α i | α P Iq for i P r , kq.Note that while the definition of -Kpk, Kq formally depends on the choice of a good κ-sequence, the structure of the graph is independent of that choice.Remark 3.2.-Kpk, K 0 q is isomorphic to a subgraph of -Kpk, K 1 q, and vice versa, for any two good κ-sequences K 0 , K 1 .
Given -Kpk, Kq as above, let S i denote the star Kptx i u, Ť Y i q for all i P r , kq.Consider the union of -Kpk, Kq with Ť iPr ,kq S i .We call this graph -F K k,κ pKq, or an -degenerate frayed K k,κ (with respect to K).As before, any vertex y i for i P r0, q is called a degenerate vertex of -F K k,κ pKq, and any Note that Remark 3.2 naturally extends to -F K k,κ pKq.Hence for each κ we now fix a specific good κ-sequence and write just -F K k,κ when talking about an -degenerate frayed K k,κ regarding that sequence.Further note that k-F K k,κ is isomorphic to K k,κ .We also call a 0-degenerate frayed K k,κ just a frayed K k,κ or F K k,κ for short, see Figure 3.3 for an example.
For a singular cardinal κ and for any P r0, ks the graph -F K k,κ is a k-typical graph with a core of size κ.These are besides the complete bipartite graph K k,κ the only other k-typical graphs with a core of size κ if κ has uncountable cofinality.
Next we will describe the other possibilities of k-typical graphs for singular cardinals with countable cofinality.and has size κ as illustrated by the bracket.
A singular k-blueprint B is a 5-tuple p , f, B, D, σq such that ‚ pB, Dq is a pk ´ ´f q-blueprint with 2 ¨|D| ď |V pBq|; and Let B " p , f, B, D, σq be a singular k-blueprint and let K " pκ α ă κ | α P ℵ 0 q be a good κ-sequence.We construct our desired graph T k pBqpKq as follows.We start with -F K k,κ pKq with the same notation as above.We remove the set tx i | i P r `f, kqu from the graph we constructed so far.Moreover, we take the disjoint union with NpB{Dq as above.
We identify the vertices ty α i | i P r `f, kqu with distinct vertices of the p2α `|V pBq|q-th and p2α `1 `|V pBq|q-th layer for every α P ℵ 0 as given by the map σ, that is where π 0 and π 1 denote the projection maps for the tuples in the image of σ.For convenience we denote a vertex originated via such an identification by any of its previous names.The For an example we refer to Figure 3.4.
As before, the information given by a specific good κ-sequence does not matter for the structure of the graph.Similarly, we get with Remark 3.2 that two graphs T k pBqpK 0 q and T k pBqpK 1 q obtained by different good κ-sequences K 0 , K 1 are isomorphic to fbs-minors of each other.Hence when we use the fixed good κ-sequence as before, we call the graph just T k pBq.Note that as before there are up to isomorphism only finitely many k-typical graphs with a core of size κ.
In summary we get for each k P N and each infinite cardinal κ a finite list of k-typical graphs with a core of size κ: Note that for the finiteness of this list we need the fixed good κ-sequence for a singular cardinal κ.
Lemma 3.4.The core of a k-typical graph is k-connected in that graph.

Generalised k-typical graphs.
The k-typical graphs cannot serve for a characterisation for the existence of k-connected sets as in Theorem 1(c) via subdivisions, as the following example illustrates.Consider two disjoint copies of the K 2,ℵ 0 together with a matching between the infinite sides, see To solve this problem we introduce generalised k-typical graphs, where we 'blow up' some of the vertices of our k-typical graph to some finite tree, e.g. an edge in the previous example.This then will allow us to obtain the desired subdivisions for our characterisation.Let G be a graph, v P V pGq be a vertex, T be a finite tree and γ : N pvq Ñ V pT q be a map.We define the pv, T, γq-blow-up of v in G as the operation where we delete v, add a vertex set tvu ˆV pT q disjointly and for each w P N pvq add the edge between w and pv, γpwqq.We call the resulting graph Gpv, T, γq.
Given blow-ups pv, T v , γ v q and pw, T w , γ w q in G, we can apply the blow-up of w in Gpv, T v , γ v q by replacing v in the preimage of γ w by pv, γ v pwqq.We call this graph Gpv, T v , γ v qpw, T w , γ w q.Note that no matter in which order we apply the blow-ups we obtain the same graph, that is Gpv, T v , γ v qpw, T w , γ w q " Gpw, T w , γ w qpv, T v , γ v q.We analogously define for a set O " tpv, T v , γ v q | v P W u of blow-ups for some W Ď V pGq the graph GpOq obtained by successively applying all the blow-ups in O.Note that if W is infinite, then GpOq is still well-defined, since each edge gets each of its endvertices modified at most once.
A type-1 k-template T 1 is a triple pT, γ, cq consisting of a finite tree T , a map γ : r0, kq Ñ V pT q and a node c P V pT q such that each node of degree 1 or 2 in T is either c or in the image of γ.Note that for each k there are only finitely many type-1 k-templates up to isomorphisms of the trees, since their trees have order at most 2k `1.
Analogously, we obtain a generalised -F K k,κ pKq for any good κ-sequence K as A type-2 k-template T 2 for a k-blueprint pB, Dq is a set tpb, P b , γ b q | b P V pBq Du of blow-ups in B such that for all b P V pBq D ‚ P b is a path of length at most k `2; ‚ the endnodes of Note that for each k there are only finitely many type-2 k-templates, up to isomorphisms of the trees in the k-blueprints and the paths for the blow-ups.
Let T 2 " tpb, T b , γ b q | b P V pBq Du be a type-2 k-template for a k-blueprint pB, Dq.
Let B " pB, D, cq be a regular k-blueprint and let T 2 " tpb, T b , γ b q | b P V pBq Du be a type-2 k-template for pB, Dq.We call T k pBqpT 2 q :" T k pBqpO 2 q a generalised T k pBq with core CpT k pBqpT 2 qq :" V pN c q ˆtv c 1 u.For an example that generalises the graph of  In grey we represent the blow-up of P as given by some type-2 k-template.
The crosses represent the core.
A type-3 k-template T 3 for a singular k-blueprint B " p , f, B, D, σq is a tuple pT 1 , T 2 q consisting of a type-1 p `f q-template T 1 and a type-2 pk ´ ´f q-template T 2 .Note that for each k there are only finitely many type-3 k-templates up to isomorphisms as discussed above for T 1 and T 2 .
Let T 3 " pT 1 , T 2 q be a type-3 k-template with T 1 " pT, γ, c 1 q for a singular k-blueprint Let O 1 2 :" tpb n , T b , γn b q | pb n , T b , γ n b q P O 2 u denote the corresponding set of blow-ups in T k pBq and let O 1  1 be for T 1 as above.The graph T k pBqpT 3 q :" T k pBqpO 1 1 Y O 1 2 q is a generalised T k pBq with core CpT k pBqpT 3 qq :" Ť Z ˆtc 1 u.For an example that generalises the graph of Figure 3.4 see Figure 3.8.We call the graph from which a generalised graph is obtained via this process its parent.
As before, Remark 3.2 and its extensions extend to generalised k-typical graphs as well.
Remark 3.5.Every -F K k,κ pT 1 qpKq or T k pBqpT 3 qpKq for a singular k-blueprint B, a type-1 k-template T 2 , a type-2 k-template T 3 and a good κ-sequence K, contains a subdivision of -F K k,κ pT 1 q or T k pBqpT 3 q respectively.
A generalised k-typical graph is either K k,κ pT 1 q, -F K k,κ pT 1 q, T k pBqpT 2 q or T k pB 1 qpT 3 q for any type-1 k-template T 1 , any P r0, kq, any regular k-blueprint B, any type-2 ktemplate T 2 for B, any singular k-blueprint B 1 and any type-3 k-template T 3 for B 1 .As with the k-typical graphs we obtain that this list is finite.
Corollary 3.6.The core of a generalised k-typical graph is k-connected in that graph.

Statement of the Main Theorem.
Now that we introduced all k-typical and generalised k-typical graphs, let us give the full statement of our main theorem.
Theorem 2. Let G be an infinite graph, let k P N, let A Ď V pGq be infinite and let κ ď |A| be an infinite cardinal.Then the following are equivalent.
(a) There is a subset (b) There is a subset A 2 Ď A with |A 2 | " κ such that there is a k-typical graph which is a minor of G with finite branch sets and with A 2 along its core.
(c) There is a subset A 3 Ď A with |A 3 | " κ such that there G contains a subdivided generalised k-typical graph with A 3 as its core.
(d) There is no nested separation system N Ď S k pGq such that every part P of N can be separated from A by less than κ vertices.

Moreover, if these statements hold, we can choose
Note that for A " V pGq we obtain the simple version as in Theorem 1 by forgetting the extra information about the core.§4.k-connected sets, minors and topological minors In this section we will collect a few basic remarks and lemmas on k-connected sets and how they interact with minors and topological minors for future references.We omit some of the trivial proofs.

Lemma 4.3. For k P N, if G contains the subdivision of a generalised k-typical graph T
with core A, then the parent of T is an fbs-minor with A along its core.
A helpful statement for the upcoming inductive constructions would be that for every vertex v of G, every large k-connected set in G contains a large subset which is pk ´1qconnected in G ´v.But while this is a true statement (cf.Corollary 8.2), an elementary proof of it seems to be elusive if v is not itself contained in the original k-connected set.
The following lemma is a simplified version of that statement and has an elementary proof.
Lemma 4.4.Let k P N and let A Ď V pGq be infinite and k-connected in G. Then for any finite set S Ď V pGq with |S| ă k there is a subset Proof.Without loss of generality we may assume that A and S are disjoint.Take a sequence pB α | α P |A|q of disjoint subsets of A with |B α | " k.For every α P |A| t0u there is at least one path from B 0 to B α disjoint from S. By the pigeonhole principle there is some v P B 0 such that |A| many of these paths start in v. Now let A 1 be the set of endvertices of these paths.§5.Structure within ends This section studies the structure within an end of a graph.
In Subsection 5.1 we will extend to arbitrary infinite graphs a well-known result for locally finite graphs relating end degree with a certain sequence of minimal separators, making use of the combined end degree.
Subsection 5.2 is dedicated to the construction of a uniformly connecting structure between disjoint rays in a common end and vertices dominating that end.

End defining sequences and combined end degree.
For an end ω P ΩpGq and a finite set S Ď V pGq let CpS, ωq denote the unique component of G ´S that contains ω-rays.A sequence pS n | n P Nq of finite vertex sets of G is called an ω-defining sequence if for all n, m P N with n ‰ m the following hold: Note that for every ω-defining sequence pS n | n P Nq and every finite set X Ď V pGq we can find an N P N such that X Ď G ´CpS N , ωq.Hence we shall also refer to the sets S n in such a sequence as separators.Given n, m P N with n ă m, let GrS n , S m s denote GrpS n Y CpS n , ωqq CpS m , ωqs, the graph between the separators.
For ends of locally finite graphs there is a characterisation of the end degree given by the existence of certain ω-defining sequences.The degree of an end ω is equal to k P N, if and only if k is the smallest integer such that there is an ω-defining sequence of sets of size k, cf.[17,Lemma 3.4.2].In this subsection we extend this characterisation to arbitrary graphs with respect to the combined degree.Recall the definition of the combined degree, ∆pωq :" degpωq `dompωq.
In arbitrary graphs ω-defining sequences need not necessarily exist, e.g. in K ℵ 1 .We start by characterising the ends admitting such a sequence.Ť tP d | d P Dompωqu is finite, we can find an N P N such that v P X Ď G ´CpS N , ωq, a contradiction.Otherwise apply Lemma 2.5 to X X Dompωq in GrXs.Note that in GrXs all vertices of X XDompωq have degree 1.Furthermore, we know that V pRqXX Ď Dompωq, since no P d contains a vertex of R as an internal vertex.But then the centre of a star would be a vertex dominating ω in X Dompωq and the spine of a comb would contain an ω-ray disjoint to R as a tail, again a contradiction.
In the proof of the end-degree characterisation via ω-defining sequences we shall need the following fact regarding the relationship of degpωq and dompωq.Lemma 5.2.If degpωq is uncountable for ω P ΩpGq, then dompωq is infinite.
Proof.Suppose for a contradiction that dompωq ă ℵ 0 .For G 1 :" G ´Dompωq let R be a set of disjoint ωaeG 1 -rays of size ℵ 1 , which exist by Remark 2.7.Let T be a transversal of tV pRq | R P Ru.Applying Lemma 2.5 to T yields a subdivided star with centre d and uncountably many leaves in T .Now d R Dompωq dominates ωaeG 1 in G 1 and hence ω in G by Remark 2.7, a contradiction.
Let ω P ΩpGq be an end with dompωq " 0, pS n | n P Nq be an ω-defining sequence and R be a set of disjoint ω-rays.We call ppS n | n P Nq, Rq a degree witnessing pair for ω, if for all n P N and for each s P S n there is a ray R P R containing s and every ray R P R meets S n at most once for every n P N. Note that this definition only makes sense for undominated ends, since a ray that contains a dominating vertex meets eventually all separators not only in that vertex.Lemma 5.3.Let ω P ΩpGq be an end with dompωq " 0. Then there is a degree witnessing pair ppS n | n P Nq, Rq.
Proof.By Lemma 5.1 and Lemma 5.2 there is an ω-defining sequence pS 1 n | n P Nq.Since ω is undominated, the separators are pairwise disjoint.
We want to construct an ω-defining sequence pS n | n P Nq with the property, that for all n P N and for all m ą n there are |S n | many S n -S m paths in GrS n , S m s.
Let S 0 be an S 1 0 -S 1 f p0q separator for some f p0q P N which is of minimum order among all candidates separating S 1 0 from S 1 m for any m P N. Suppose we already constructed the sequence up to S n .Let S n`1 be an S 1 f pnq`1 -S 1 f pn`1q separator for some f pn `1q ą f pnq `1 which is of minimum order among all candidates separating S 1 f pnq`1 and S 1 m for any m ą f pnq `1.We can easily lift these results to ends dominated by finitely many vertices with the following observation based on Remark 2.7.
Finally, we state more remarks on the relationship between degpωq and dompωq similar to Lemma 5.2 without giving the proof.
(1) If dompωq is infinite, then so is degpωq for every ω P ΩpGq.
(3) There is a graph with an end ω 1 such that degpω 1 q " κ 1 and dompω 1 q " κ 2 , namely the union of the complete bipartite graph KpA, Bq with |A| " κ 1 , |B| " κ 2 with the complete graph on A.
(4) There is a graph with an end ω 1 such that degpω 1 q " k 1 and dompω 1 q " k 2 .

Constructing uniformly connected rays.
Let ω P ΩpGq be an end of G and let I, J be disjoint finite sets with 1 ď |I| ď degpωq and 0 ď |J| ď dompωq.Let R " pR i | i P Iq be a family of disjoint ω-rays and let isomorphic to a subdivision of T pT 2 q.Moreover, the subdivision of v i K P i v i J is the segment R i X Γ for all i P I such that v i J corresponds to the top vertex of that segment.Then pR, Dq is called pT, T 2 q-connected if for every finite X Ď V pGq D there is a pT, T 2 q-connection avoiding X. Lemma 5.10.Let ω P ΩpGq, let R " pR i | i P Iq be a finite family of disjoint ω-rays with |I| ě 1 and let D " pd j P Dompωq | j P Jq be a finite family of distinct vertices disjoint from Ť R with I X J " ∅.Then there is a tree T on I Y J and a simple type-2 |I Y J|-template T for pT, Jq such that pR, Dq is pT, T q-connected.
Proof.Let X Ď V pGq D be any finite set.We extend X to a finite superset X 1 such that R i X X 1 is an initial segment of R i for each i P I, and such that D Ď X 1 .As all rays in R are ω-rays, we can find finitely many Ť R -Ť R paths avoiding X 1 which are internally disjoint such that their union with Ť R is a connected subgraph of G. Moreover it is possible to do this with a set P of |I| ´1 many such paths in a tree-like way, i.e. contracting a large enough finite segment avoiding X 1 of each ray in R and deleting the rest yields a subdivision Γ 1 X of a tree on I whose edges correspond to the paths in P. For each vertex d j we can moreover find a d j -Ť R path avoiding V pΓ 1 X q Y X 1 td j u and all paths we fixed so far.This yields a tree T X on I Y J and a simple type-2 k-template T X for pT X , Jq such that J is a set of leaves and a pT X , T X q-connection Γ X avoiding X.Now we iteratively apply this construction to find a family pΓ i | i P Nq of pT i , T i qconnections such that Γ m ´D and Γ n ´D are disjoint for all m, n P N with m ‰ n.By the pigeonhole principle we now find a tree T on I Y J, a type-2 |I Y J|-template T and an infinite subset N Ď N such that pT n , T n q " pT, T q for all n P N .Now for each finite set X Ď V pGq D there is an n P N such that Γ n and X are disjoint, hence pR, Dq is pT, T q-connected.Proof.By Lemma 5.10 there is a tree T and a simple type-2 |I Y J|-template T such that pR, Dq is pT, T q-connected.Let pΓ i | i P Nq be a family of pT, T q-connections such that Γ m ´D and Γ n ´D are disjoint for all m, n P N with m ‰ n.
Finally, this result can be lifted to the minor setting by Lemma 4.3.
Corollary 5.12.Let ω P ΩpGq, let R " pR i | i P Iq be a finite family of disjoint ω-rays with |I| ě 1 and let D " pd j P Dompωq | j P Jq be a finite family of distinct vertices disjoint from Ť R with I X J " ∅.Then there is a tree T such that G contains NpT {Jq as an fbs-minor.§6.Minors for regular cardinalities This section is dedicated to prove the equivalence of (a), (b) and (c) of Theorem 2 for regular cardinals κ.

Complete bipartite minors.
In this subsection we construct the complete bipartite graph K k,κ as the desired minor (and a generalised version as the desired subdivision), if possible.The ideas of this construction differ significantly from Halin's construction [11,Thm. 9.1] of a subdivision of K k,κ in a k-connected graph of uncountable and regular order κ.‚ or there is no end in the closure of A; ‚ or there is an end ω in the closure of A with dompωq ě k; then there is a subset along its core.
Moreover, the branch sets for the vertices of the finite side of K k,κ are singletons.
Proof.We iteratively construct a sequence of subgraphs H i for i P r0, kq witnessing that K i,κ is a minor of G. Furthermore, we incorporate that the branch sets for the vertices of the finite side of K i,κ are singletons tv j | j P r0, iqu and the branch sets for the vertices of the infinite side induce finite trees on H i each containing a vertex of A. Moreover, we will guarantee the existence of a subset , iqu and such that each vertex of A i is contained in a branch set of H i and each branch set of H i contains precisely one vertex of A i .
Set G 0 :" G, A 0 :" A and H 0 " GrAs.For any i P r0, kq we inductively apply Lemma 2.5 (and in the third case also Remark 2.6) to A i in G i to find a subdivided star S i with centre v i and κ many leaves L i Ď A i .Without loss of generality we can assume v i R V pH i q, since otherwise we could just remove the branch set containing v i and from A i the vertex contained in that branch set.Moreover, by Lemma 4.4 we find a subset . First we remove from H i every branch set which corresponds to a vertex of the infinite side of K i,κ and does not contain a vertex of L 1 i .Now each path in S i from a neighbour of v i to L i eventually hits a vertex of one of the finite trees induced by one of the remaining branch sets of H i .Since all these paths are disjoint, only finitely many of them meet the same branch set first.Thus κ many different of the remaining branch sets are met by those paths first.To get H i`1 we do the following.
First we add tv i u as a new branch set.Then each of the κ many branch sets reached first as described above we extend by the path segment between v i and that branch set of precisely one of those paths.Finally, we delete all remaining branch sets not connected to tv i u.With A i`1 :" L 1 i X V pH i`1 q we now have all the desired properties.
Finally, setting H :" H k and A 1 :" A k finishes the construction.
Let H be an inflated subgraph witnessing that K k,κ " Kpr0, kq, Zq is an fbs-minor of G with A along Z for some A Ď V pGq where each branch set of x P r0, kq is a singleton.Given a type-1 k-template T 1 " pT, γ, cq we say H is T 1 -regular if for each z P Z: ‚ there is an isomorphism ϕ z : T 1 z Ñ T z between a subdivision T 1 z of T and the finite tree T z " HrBpzqs; ‚ xϕ z pγpxqq P EpHq for each x P r0, kq; and We say G contains K k,κ as a T 1 -regular fbs-minor with A along Z if there is such a T 1 -regular H. Lemma 6.2.Let k P N and κ be a regular cardinal.If K k,κ is an fbs-minor of G with A 1 along its core where each branch set of x P r0, kq is a singleton, then there is type- Proof.Let H be the inflated subgraph witnessing that K k,κ is an fbs-minor as in the statement.Let x also denote the vertex of G in the branch set Bpxq of x P r0, kq.Let v z x P Bpzq denote the unique endvertex in Bpzq of the edge corresponding to xz P EpM q (cf.Section 2).Let T z denote a subtree of HrBpzqs containing B z " tv z x | x P r0, kqu Y ta z u for the unique vertex a z P A X Bpzq.Without loss of generality assume that each leaf of T z is in B z .By suppressing each degree 2 node of T z that is not in B z , we obtain a tree suitable for a type-1 k-template where a z is the node in the third component of the template.
By applying the pigeonhole principle multiple times there is a tree T such that there exist an isomorphism ϕ z : T 1 z Ñ T z for a subdivision T 1 z of T for all z P Z 1 for some Z 1 Ď Z with |Z 1 | " κ, such that tϕ z pv z x q | z P Z 1 u is a singleton tt x u for all x P r0, kq as well as tϕ z pa z q | z P Z 1 u is a singleton tcu.
Therefore with γ : r0, kq Ñ V pT q defined by x Þ Ñ t x and c defined as above, we obtain a type-1 k-template T 1 :" pT, γ, cq such that the subgraph H 1 of H where we delete each branch set for z P Z Z 1 is T 1 -regular.
Hence, we also obtain a subdivision of a generalised K k,κ .
Corollary 6.3.In the situation of Lemma 6.1, there is contains a subdivision of a generalised K k,κ with core A 2 .

Minors for regular k-blueprints.
In this subsection we construct the k-typical minors for regular k-blueprints, if possible.
While these graphs are essentially the same minors given by Oporowski, Oxley and Thomas [15,Thm. 5.2], we give our own independent proof based on the existence of an infinite k-connected set instead of the graph being essentially k-connected.
The first lemma constructs such a graph along some end of high combined degree.The following lemma allows us to apply Lemma 6.4 when Lemma 6.1 is not applicable.We close this subsection with a corollary that is not needed in this paper, but provides a converse for Lemma 6.5 as an interesting observation.
Corollary 6.6.Let ω P ΩpGq be an end of G with ∆pωq ě k P N. Then every subset Proof.By Lemma 6.4 we obtain a subdivision of a generalised T k pBq with core A 1 for some (a) There is a subset  In this subsection, given a k-connected set A of size κ, we will construct an -Kpk, Kq minor in G for some suitable P r0, ks and good κ-sequence K with a suitable subset of A along its precore.This minor is needed as an ingredient for any of the possible k-typical graphs but the K k,κ (which we obtain from the following lemma if " k).Let A " pA α Ď A | α P cf κq be a family of disjoint subsets of A. We say that G contains -Kpk, Kq as an fbs-minor with A along its precore Z if the map mapping each vertex of the inflated subgraph to its branch set induces a bijection between A α and Z α for all α P cf κ.Proof.We start with any good κ-sequence K " pκ α ă κ | α P cf κq.We construct the desired inflated subgraph by iteratively applying Lemma 6.1.
For α P cf κ suppose we have already constructed for each β ă α an inflated subgraph H β witnessing that K k,κ β is an fbs-minor of G with some A β Ď A along its core.Furthermore, suppose that the branch sets of the vertices of the finite side are singletons and the branch sets of the vertices of the infinite side are disjoint to all branch sets of H γ for all γ ă β.
We apply Lemma 6.1 for κ α to any set A 1 Ď A Ť βăα A β of size κ α to obtain an inflated subgraph for K k,κα with the properties as stated in that lemma.If any branch set for a vertex of the infinite side meets any branch set we have constructed so far, we delete it.
Since κ α is regular and κ α ą cf κ, the union of all inflated subgraphs we constructed so far has order less than κ α .We obtain that the new inflated subgraph (after the deletions) still witnesses that K k,κα is an fbs-minor of G with some A α Ď A 1 along its core.If a branch set for the finite side meets any branch set of a vertex for the infinite side for some β ă α, we delete that branch set and modify A β accordingly.As the union of all branch sets for the finite side we will construct in this process has cardinality cf κ, each A β will loose at most cf κ ă κ β many elements, hence will remain at size κ β for all β P cf κ.We denote the sequence pA α | α P cf κq with A.
By Lemma 2.9 there is an ď k and an I Ď cf κ with |I| " cf κ such that H α and H β have precisely branch sets for the vertices of the finite side in common for all α, β P I.
Hence relabelling the subsequences KaeI and AaeI to KaeI and AaeI respectively as discussed in Section 3 yields the claim, where the union of the respective subgraphs H α is the witnessing inflated subgraph.

Frayed complete bipartite minors.
In this subsection we will construct a frayed complete bipartite minor, if possible.We shall use an increasing amount of fixed notation in this subsection based on Lemma 7.1, which we will fix as the situation in which we continue our construction.Let H be an inflated subgraph witnessing that G contains -Kpk, Kq as an fbs-minor with A along Z as in Lemma 7.1.To simplify our notation, we denote the unique vertex of H in a branch set of y α i also by y α i for all α P cf κ and i ă k.Similarly, we denote the set ty α i P V pHq | i P r0, kqu also with Y α for all α P cf κ, and denote the family pY α Ď V pHq | α P cf κq with Y.Moreover, let H α denote the subgraph of H witnessing that KpY α , Z α q is an fbs-minor of G with A α along Z α .Finally, let D " ty i | i P r0, qu " Ş tV pH α q | α P cf κu denote the set of degenerate vertices of -Kpk, Kq.
For a set U Ď V pGq and α P cf κ, a Y α -U bundle P α is the union Ť tP α i | i P r0, kqu of k disjoint paths, where P α i Ď G is a (possibly trivial) Y α -U path starting in y α i P Y α and ending in some u α i P U .A family P " pP α | α P cf κq of Y α -U bundles is a Y -U bundle if P α ´U and P β ´U are disjoint for all α, β P cf κ with α ‰ β.Note that if a Y -U bundle exists, then U contains D .
A set U Ď V pGq distinguishes Y if whenever y α i and y β j are in the same component of G ´U for α, β P cf κ and i, j P r0, kq, then α " β.For a cardinal λ, a set W Ď V pGq is λ-linked to a set U Ď V pGq, if for every w P W and every u P U there are λ many internally disjoint wu paths in G.
The following lemma is the main part of the construction.Lemma 7.4.In Situation 7.2, suppose there is a set U Ď V pGq such that ‚ there is a Y -U bundle P " pP α | α P cf κq; and Then there is an I 0 Ď cf κ with |I 0 | " cf κ and a family A 0 " pA α 0 Ď A α | α P I 0 q with |A α 0 | " κ α for all α P I 0 such that -F K k,κ pKaeI 0 q is an fbs-minor of G with A 0 along ZaeI 0 .
Proof.Let U , P and W be as above.By Lemma 2.9 there is a j P r0, ks and a subset I 1 Ď cf κ with |I 1 | " cf κ such that (after possibly relabelling the sets Y α for all α P I 1 simultaneously) for every α, β P I 1 with α ‰ β ‚ y i " u α i " u β i for all i P r0, q; ‚ x i :" u α i " u β i for all i P r , `jq; and ‚ u α i 0 ‰ u β i 1 for all i 0 , i 1 P r `j, kq.Furthermore, after deleting at most j more elements from I 1 we obtain I 2 such that ‚ u α i ‰ y α i for all i P r , `jq and all α P I 2 .Note that if |U | ă cf κ, then `j " k and we set I 0 :" I 2 and L :" ∅.
Otherwise we construct subdivided stars with distinct centres in W .We start with a k ´ ´j element subset W 1 " tw i | i P r `j, kqu Ď W disjoint from both D as well such that L is the disjoint union of subdivided stars S i for all i P r `j, kq with centre w i and leaves u α i , and L is disjoint to P α ´tu α i | i P r `j, kqu for all α P IpLq.A partial star-link L is a star-link if |IpLq| " cf κ.Note that the union of a chain of partial star-links (ordered by the subgraph relation) yields another partial star-link.Hence by Zorn's Lemma there is a maximal partial star-link M .Assume for a contradiction that M is not a star-link.
Then the set N " V pM q Y Ť αPIpM q V pP α q has size less than cf κ.Take some β P I 1 IpM q such that M is disjoint to P β .Since W is cf κ-linked to U , we can find k ´ ´j disjoint W 1tu β i | i P r `j, kqu paths disjoint from N W 1 , contradicting the maximality of M (after possibly relabelling).Hence there is a star-link L, and we set I 0 :" IpLq.
Let H I 0 denote the subgraph of H containing only the branch sets for vertices in Y α Y Z α for α P I 0 .Since L Y Ť αPI 0 P α has size cf κ ă κ α for all α P I 0 , we can remove every branch set for some z P Z α meeting L Y Ť αPI 0 P α and obtain A α 0 Ď A α with |A α 0 | " κ α .The union of the resulting subgraph with L and Ť αPI 0 P α witnesses that -F K k,κ pKaeI 0 q is an fbs-minor of G with A 0 :" pA α 0 | α P I 0 q along ZaeI 0 .
As before, the previous lemma can be translated to find a desired subdivision of a generalised -F K k,κ .
Lemma 7.5.In the situation of Lemma 7.4, there is an The next lemma reroutes some rays to find a bundle from Y n to those new rays and dominating vertices with some specific properties.
Lemma 7.16.In Situation 7.9, there is a set R Hence we include references to these objects into Situation 7.9.
Proof.Given n P N, let P n be as in Corollary 7.13.Let P be a set of paths in G 1 rS n , S n`1 s each with end vertices R 1 X pS n Y S n`1 q for some R 1 P R 1 .We call such a set P feasible.For a feasible P, let P n pPq denote the Y n -pDompωq Y Ť Pq bundle contained in P n and let p n pPq denote the finite parameter |pP n ´P n pPqq ´Ť P|.
Note that tR 1 X G 1 rS n , S n`1 s | R 1 P R 1 u is a feasible set.Now choose a feasible P n such that p n pP n q is minimal and let Q n :" P n pP n q.
Assume for a contradiction that there is a path and v 2 denote vertices in this intersection such that v 1 P V pv 0 P v 2 q.We replace the segment v 0 P v 2 by the path consisting of the paths Q n i and Q n j that contain v 0 and v 2 respectively, as well as any y n iy n j path in H n avoiding the finite set Dompωq Y Q n Y S n Y S n`1 .The resulting set P is again feasible and the parameter p n pPq is strictly smaller than p n pP n q, contradicting the choice of P n .Now let R 2 be the set of components in the union Ť tP n | n P Nu.Indeed, this is a set of ω 1 -rays that together with the bundles Q n satisfy the desired properties.
For m, n P N, we say Q m and Q n follow the same pattern, if for all i, j P r0, kq ‚ Q m i and Q n i either meet the same ray in R 2 or the same vertex in Dompωq; ‚ if Q m i and Q m j both meet some R P R 2 and Q m i meets R closer to the start vertex of R than Q m j , then Q n i meets R closer to the start vertex of R than Q n j .
Lemma 7.17.In Situation 7.9, we may assume without loss of generality that ‚ there are integers k 0 , k 1 , f P N with 1 ď k 0 ď degpω 1 q, 0 ď `f `k1 ď dompωq and   ‚ it is a degenerate vertex or frayed centre of -F K k,κ if T " -F K k,κ for some P r0, ks; or ‚ it is a dominating vertex, a degenerate vertex or a frayed centre of T k pBq if T " T k pBq for some regular or singular k-blueprint B.
We distinguish four cases.
If v R V pHq, then H Ď G ´v still witnesses that T is an fbs-minor of G ´v with A 1 :" A 2 along its core.
If v belongs to a branch set of a vertex c of the core, then the inflated subgraph obtained by deleting that branch set still yields a witness that T is an fbs-minor of G ´v with A 1 :" A 2 tcu along its core.
If v belongs to a branch set of an essential vertex w P V pT q, then the inflated subgraph where we delete this branch set from H witnesses that the obvious pk ´1q-typical subgraph of T ´w is an fbs-minor of G ´v with A 1 :" A 2 along its core.
If v belongs to a branch set of a vertex w P V pNpB{Dq ´Dq, then we delete the branch sets of the layers (not including D) up to the layer containing w and relabelling accordingly (and modifying the κ-sequence if necessary).This yields a supergraph of an inflated subgraph witnessing that T is an fbs-minor of G ´v with A 1 along its core for some A 1 Ď A 2 with |A 1 | " κ.Similar arguments yield the statement if v belongs to a branch set of a neighbour of a frayed centre.
In any case, with the other direction of Theorem 6.7 or Theorem 7.19 we get that A 1 is pk ´1q-connected in G ´v.
As another corollary we prove that we are able to find k-connected sets of size κ in sets which cannot be separated by less than κ many vertices from another k-connected set.This will be an important tool for our last part of the characterisation in the main theorem.Proof.Let P be a set of κ many disjoint A -B paths as given by Theorem 2.2.Let B 1 denote B X Ť P. Let H Ď G be an inflated subgraph witnessing that a k-typical graph is an fbs-minor of G with B 2 along its core for some B 2 Ď B 1 with |B 2 | " κ as given by Theorem 6.7 or Theorem 7.19.Let P 1 denote the set of the A -H subpaths of the A -B 2 paths in P. We distinguish two cases.
If the k-typical graph is a T k pBq for some regular k-blueprint B " pT, D, cq, then (since each branch set in H is finite) there is an infinite subset P 2 Ď P 1 and a node c 1 P V pT Dq such that each branch set in H of vertices in V pN c 1 q meets Ť P 2 at most once and no other branch set meets Ť P 2 .Let A 1 :" Ť P 2 X A. We extend each of these branch sets with the path from P 2 meeting it.This yields a subgraph H 1 witnessing that T k pT, D, c 1 q is an fbs-minor of G with some A 2 along its core with A 1 Ď A 2 .
Otherwise, since each branch set in H is finite, there is a subset P 2 Ď P 1 of size κ such that each branch set in H of vertices corresponding to the core meets Ť P 2 at most once and no other branch set meets Ť P 2 .Let A 1 :" Ť P 2 X A. Again, we extend each of these branch sets with the path from P 2 meeting it.This yields a subgraph H 1 witnessing that the same k-typical graph is an fbs-minor of G with some A 2 along its core such that Applying Theorem 6.7 or Theorem 7.19 again together with Remark 4.1 yields the claim.§9.Nested separation systems This section will finish the proof of Theorem 2 by proving the duality theorem and hence providing the last equivalence of the characterisation.
Recall that a nested separation system N Ď S k pGq is called k-lean if given any two (not necessarily distinct) parts P 1 , P 2 of N and vertex sets Z 1 Ď P 1 , Z 2 Ď P 2 with

Lemma 2 . 5 (
Star-Comb Lemma).Let U Ď V pGq be infinite and let κ ď |U | be a regular cardinal.Then the following are equivalent.
κq a family of finite subsets of U .Then there is a finite set D Ď U and a set I Ď κ with |I| " κ such that F α X F β " D for all α, β P I with α ‰ β.A separation of G is a tuple pA, Bq of vertex sets such that A Y B " V pGq and such that there is no edge of G between A B and B A. The set A X B is the separator of pA, Bq and the cardinality |A X B| is called the order of pA, Bq.Given k P N, let S k pGq denote the set of all separations of G of order less than k.Two separations pA, Bq and pC, Dq are nested if either A Ď C and D Ď B, or A Ď D and C Ď B holds.A set N of separations of G is called a nested separation system of G if it is symmetric, i.e. pB, Aq P N for each pA, Bq P N and nested, i.e. the separations in N are pairwise nested.An orientation O of a nested separation system N is a subset of N that contains precisely one of pA, Bq and pB, Aq for all pA, Bq P N .An orientation O of N is consistent if whenever pA, Bq P O and pC, Dq P N with C Ď A and B Ď D, then pC, Dq P O.For each consistent orientation O of N we define a part P O of N as the vertex set Ş tB | pA, Bq P Ou.

κFigure 3 . 1 .Lemma 3 . 1 .
Figure 3.1.A stylised version of a K 4,κ , where the large box stands for the core of κ many vertices and the dashed lines from a vertex to the corners of the box represent that this vertex is connected to all vertices in the box.

Figure 3 . 2 .
Figure 3.2.Image of T 4 pP, tdu, cq where P " cabd is a path of length 3 between nodes c and d.P 0 is represented in gray.The crosses represent its core.

Figure 3 . 3 .
Figure 3.3.Image of 2 -F K 4,κ .The black squares represent the frayed centres and the white squares the degenerate vertices.Its core is represented by the union of the boxes (labelled according to the fixed good κ-sequence)

Figure 3 . 5 .
Figure 3.5.Now the vertices of the infinite side from one of the copies is a 4-connected set in that graph, but the graph does not contain any subdivision of a 4-typical graph, since it neither contains a path of length greater than 13 (and hence no subdivision of a T 4 pBq for some regular k-blueprint B), nor a subdivision of a K 4,ℵ 0 .

Figure 3 . 5 .
Figure 3.5.A graph with an infinite 4-connected set (marked by the cross vertices) containing no subdivision of a 4-typical graph.

Figure 3 . 6 .
Figure 3.6.Image of a generalised K 4,κ on the left.The crosses represent the core.On the right is how we represent the same graph in a simplified way by labelling the vertices according to their adjacencies.

Figure 3 . 7 .
Figure 3.7.Image of a generalised T 4 pP, tdu, cq for P , tdu, c as in Figure 3.2.In grey we represent the blow-up of P as given by some type-2 k-template.

1 if
, B, D, σq.Then for pb n , T b , γ n b q P O 2 we extend γ n b v P ty n i | i P r `f, kqu and n even; v b 0 if v P ty n i | i P r `f, kqu and n odd; γ n b pvq otherwise.

Corollary 5 . 7 .Corollary 5 . 8 .
(a) For every ωaeG 1 -defining sequence pS 1 n | n P Nq of G 1 there is an ω-defining sequence pS n | n P Nq of G with S 1 n " S n Dompωq for all n P N. (b) For every ω-defining sequence pS n | n P Nq of G there is an ωaeG 1 -defining sequence pS 1 n | n P Nq of G 1 with S 1 n " S n Dompωq for all n P N. Let k P N and let ω P ΩpGq.Then ∆pωq ě k if and only if for every ω-defining sequence pS n | n P Nq the sets S n eventually have size at least k.Proof.As noted before, each vertex dominating ω has to be in eventually all sets of an ω-defining sequence.Suppose ∆pωq ě k.If dompωq ě ℵ 1 , then there is no ω-defining sequence and there is nothing to show.If dompωq " ℵ 0 , then the sets of any ω-defining sequence eventually have all size at least k.If dompωq ă ℵ 0 , we can delete Dompωq and apply Corollary 5.4 to G ´Dompωq with k 1 " degpωq.With Remark 5.6 (b) the claim follows.If ∆pωq ă k, we can delete Dompωq and apply Corollary 5.4 with k 1 " degpωq.With Remark 5.6 (a) the claim follows.Let k P N and let ω P ΩpGq.Then ∆pωq " k if and only if k is the smallest integer such that there is an ω-defining sequence pS n | n P Nq with |S n | " k for all n P N.

Corollary 5 . 11 .
Let ω P ΩpGq, let R " pR i | i P Iq be a finite family of disjoint ω-rays with |I| ě 1 and let D " pd j P Dompωq | j P Jq be a finite family of distinct vertices disjoint from Ť R with I X J " ∅.Then there is a tree T such that G contains a subdivision of a generalised NpT {Jq.

Lemma 6 . 1 .
Let k P N, let A Ď V pGq be infinite and k-connected in G and let κ ď |A| be a regular cardinal.If ‚ either κ is uncountable;

Lemma 6 . 4 .
Let ω P ΩpGq be an end of G with ∆pωq ě k P N. Let A Ď V pGq be a set with ω in its closure.Then there is a countable subset A 1 Ď A and a regular k-blueprint B such that G contains a subdivision of a generalised T k pBq with core A 1 .Proof.Let I, J be disjoint sets with |I Y J| " k and |I| ě 1.Let R " pR i | i P Iq be a family of disjoint ω-rays and D " pd j P Dompωq | j P Jq be a family of distinct vertices disjoint from Ť R. Applying Lemma 5.10 yields a tree B on I Y J and a type-2 ktemplate T for pB, Jq such that pR, Dq is pB, T q-connected.Let pΓ i | i P Nq denote the family of pB, T q-connections as in the proof of Lemma 5.10.Moreover, there is an infinite set of disjoint A -Ť R paths by Theorem 2.2 since ω is in the closure of A. Now any infinite set of disjoint A -Ť R paths has infinitely many endvertices on one ray R c for some c P I. Let A 2 denote the endvertices in A of such an infinite path system.Next we extend for infinitely many Γ i the segment of R c that it contains so that it has the endvertex of such an A 2 -R c path as its top vertex and add that segment together with the path to Γ i , while keeping them disjoint but for D. Let A 1 denote the set of those endvertices of the paths in A 2 we used to extend Γ i for those infinitely many i P N. Finally, we modifying the type-2 k-template accordingly.We obtain the subdivision of the generalised T k pB, J, cq as in the proof of Corollary 5.11.

Lemma 6 . 5 .
Let k P N, let A Ď V pGq be infinite and k-connected in G and let ω P ΩpGq be an end in the closure of A. Then ∆pωq ě k.Proof.We may assume that ∆pGq is finite.Hence without loss of generality A does not contain any vertices dominating ω.Let pS n | n P Nq be an ω-defining sequence, which exists by Lemma 5.1.Take N P N such that there is a set B Ď A CpS N , ωq of size k.For every n ą N let C n Ď A X CpS n , ωq be a set of size k, which exists since ω is in the closure of A. Since A is k-connected in G, there are k disjoint B -C n paths in G, each of which contains at least one vertex of S n .Hence for all n ą N we have |S n | ě k and by Corollary 5.7 we have ∆pωq ě k.

6 . 3 .Theorem 6 . 7 .
for a regular k-blueprint B. Corollary 3.6 yields the claim.Characterisation for regular cardinals.Now we have developed all the necessary tools to prove the minor and topological minor part of the characterisation in Theorem 2 for regular cardinals.Let G be a graph, let k P N, let A Ď V pGq be infinite and let κ ď |A| be a regular cardinal.Then the following are equivalent.
an fbs-minor of G with A 2 along its core; ‚ or T k pBq is an fbs-minor of G with A 2 along its core for some regular kblueprint B.(c) There is a subsetA 3 Ď A with |A 3 | " κ such that ‚ either G contains a subdivision of a generalised K k,κ with core A 3 ;‚ or G contains the subdivision of a generalised T k pBq with core A 3 for some regular k-blueprint B.Moreover, if these statements hold, we can chooseA 1 " A 2 " A 3 .Proof.If (b) holds, then A 2 is k-connected in Gby Lemma 4.2 with Lemma 3.4.If (a) holds, then we can find a subset A 3 Ď A 1 with |A 3 | " κ yielding (c) by either Lemma 6.1 and Corollary 6.3 or by Lemma 6.5 and Lemma 6.4.If (c) holds, then so does (b) by Lemma 4.3 with A 2 :" A 3 .Moreover, A 3 is a candidate for both A 2 and A 1 .§7. Minors for singular cardinalities In this section we will prove the equivalence of (a), (b) and (c) of Theorem 2 for singular cardinals κ.

Lemma 7 . 1 .
Let k P N, let A Ď V pGq be infinite and k-connected in G and let κ ď |A| be a singular cardinal.Then there is an P r0, ks, a good κ-sequence K " pκ α ă κ | α P cf κq, and a family A " pA α Ď A | α P cf κq of pairwise disjoint subsets of A with |A α | " κ α such that G contains -Kpk, Kq as an fbs-minor with A along Z.Moreover, the branch sets for the vertices in Ť Y are singletons.

Situation 7 . 2 .
Let k P N, let A Ď V pGq be infinite and k-connected in G and let κ ď |A| be a singular cardinal.Let ď k and let ‚ K " pκ α ă κ | α P cf κq be a good κ-sequence; and ‚ A " pA α | α P cf κq be a family of pairwise disjoint subsets of A with |A α | " κ α .
in G by Theorem 2.1.Hence we fix the initial Y α -U segments of these paths for each α P cf κ, which are disjoint outside of U by the assumption that U distinguishes Y.This yields the desired Y -U bundle.

6 or
`f `k0 `k1 " k;‚ there is a subset R 0 Ď R 2 with |R 0 | " k 0 ; and‚ there are disjoint D f , D 1 Ď Dompωq D with |D f | " f and |D 1 | " k 1 ;such that for all m, n P N(a) Q n is a Y np Ť R 0 Y D Y D f q bundle; (b) Q n X Dompωq " D f Y D ; and(a) There is a subsetA 1 Ď A with |A 1 | " κ such that A 1 is k-connected in G.(b) There is a subset A 2 Ď A with |A 2 | " κ such that‚ either G contains an -degenerate frayed K k,κ as an fbs-minor with A 2 along its core for some 0 ď ď k;‚ or T k pBq is an fbs-minor of G with A 2 along its core for a singular k-blueprint B. (c) There is a subsetA 3 Ď A with |A 3 | " κ such that ‚ either G contains a subdivision of a generalised -F K k,κ with core A 3 for some 0 ď ď k;‚ or G contains the subdivision of a generalised T k pBq with core A 3 for some singular k-blueprint B.Moreover, if these statements hold, we can chooseA 1 " A 2 " A 3 .Proof.If (b) holds, then A 2 is k-connected in Gby Lemma 4.2 with Lemma 3.4.Suppose (a) holds.Either we can find a subset A 3 Ď A 1 with |A 3 | " κ and a subdivision of -F K k,κ pKq with core A 3 for some good κ-sequence K by Lemma 7.4 and either Lemma 7.Corollary 7.8.Otherwise, we can apply Lemma 7.18 to obtain A 3 Ď A 1 with |A 3 | " κ and a subdivision of T k pBqpKq with core A 3 for some singular k-blueprint B and a good κ-sequence K.With Remark 3.5 we obtain the subdivision of the respective generalised k-typical graph with respect to the fixed good κ-sequence.If (c) holds, then so does (b) by Lemma 4.3 with A 2 :" A 3 .Moreover, A 3 is a candidate for both A 2 and A 1 .§8. Applications of the minor-characterisation In this section we will present some applications of the minor-characterisation of kconnected sets.As a first corollary we just restate the theorem for k " 1, giving us a version of the Star-Comb Lemma for singular cardinalities.For this, given a singular cardinal κ, we call the graph F K 1,κ a frayed star, whose centre is the vertex x 0 of degree cf κ and whose leaves are the vertices Ť Z.Moreover, we call the graph T 1 p0, 0, ptcu, ∅q, ∅, 0 Þ Ñ pc, 0qq a frayed comb with spine N c and teeth Ť Z.Note that each generalised frayed star or generalised frayed comb contains a subdivision of the frayed star or frayed comb respectively.Corollary 8.1 (Frayed-Star-Comb Lemma).Let U Ď V pGq be infinite and let κ ď |U | be a singular cardinal.Then the following are equivalent.(a)There is a subset U 1 Ď U with |U 1 | " κ such that U 1 is 1-connected in G.

Corollary 8 . 3 .
Let k P N, let A, B Ď V pGq be infinite and let κ ď |A| be an infinite cardinal.If B is k-connected in G and A cannot be separated from B by less than κ vertices, then there is anA 1 Ď A with |A 1 | " κ which is k-connected in G.
Let κ be a cardinal and let A, B Ď V pGq.If A and B cannot be separated by less than κ vertices, then G contains κ disjoint A -B paths.(orDrespectively).If v and w are the endvertices of P , then we denote P also by vRw (or vDw respectively).If v is the end vertex of vRw whose distance is closer to the start vertex of R, then v is called the bottom vertex of vRw and w is called the top vertex of vRw.If additionally v is the start vertex of R, then we call vRw an initial segment of R and denote it by Rw.Recall that an end of G is an equivalence class of rays, where two rays are equivalent if they cannot be separated by deleting finitely many vertices of G.We denote the set of ends of G by ΩpGq.A ray being an element of an end ω P ΩpGq is called an ω-ray.A double ray all whose tails are elements of ω is called an ω-double ray.For an end ω P ΩpGq let degpωq denote the degree of ω, that is the supremum of the set t|R| | R is a set of disjoint ω-raysu.Note for each end ω there is in fact a set R of vertex disjoint ω-rays with |R| " degpωq [12,Satz 1].
vertices such that A S and B S lie in different components of G ´S.We also say S separates A and B. For convenience, by There is a subset U 2 Ď U with |U 2 | " κ such that G either contains a subdivided star whose set of leaves is U 2 or a comb whose set of teeth is U 2 .
CpS n`1 , ωq path meets S n`1 .While for any vertex v P CpS n`1 , ωq CpS n , ωq there is a path to CpS n , ωq X CpS n`1 , ωq in CpS n`1 , ωq, this path would meet a vertex w P S n .Note that every v -R path has to contain vertices from infinitely many S n , hence it has to contain a vertex dominating ω.For each d P Dompωq let P d be either the vertex set of a v -Dompωq path containing d if it exists, or P d " ∅ otherwise.If Lemma 5.1.Let ω P ΩpGq be an end.Then there is an ω-defining sequence pS n | n P Nq if and only if ∆pωq ď ℵ 0 .Proof.Note that for all finite S Ď V pGq, no d P Dompωq can lie in a component C ‰ CpS, ωq of G ´S. Hence for every ω-defining sequence pS n | n P Nq and every d P Dompωq there is an N P N such that d P S m for all m ě N .Therefore, if dompωq ą ℵ 0 , no ω-defining sequence can exist, since the union of the separators is at most countable.Moreover, note that for every ω-defining sequence every ω-ray meets infinitely many distinct separators.It follows that degpωq is at most countable as well if an ω-defining sequence exist.For the converse, suppose ∆pωq ď ℵ 0 .Let td n | n ă dompωqu be an enumeration of Dompωq.Let R " r 0 r 1 ...be an ω-devouring ray, which exist by Lemma 2.8.We build our desired ω-defining sequence pS n | n P Nq inductively.Set S 0 :" tr 0 u.For n P N suppose S n is already constructed as desired.Take a maximal set P n of pairwise disjoint N pS n Dompωqq -R paths in CpS n , ωq.Note that P n is finite since otherwise by the pigeonhole principle we would get a vertex v P S n Dompωq dominating ω.Furthermore, P n is not empty as CpS n , ωq is connected.DefineS n`1 :" pS n X Dompωqq Y ď P n Y r m | m is minimal with r m P CpS n , ωq ( Y d m | mis minimal with d m P CpS n , ωq ( .By construction, S n`1 X S i contains only vertices dominating ω for i ď n.Let P be any S n -CpS n`1 , ωq path.We can extend P in CpS n`1 , ωq to a S n -R path.And since CpS n`1 , ωq X S n`1 is empty, we obtain P X S n`1 ‰ ∅ by construction of S n`1 .Hence any S nn -CpS n`1 , ωq path avoiding S n`1 , and hence contradicting the existence of such v. Hence CpS n`1 , ωq Ď CpS n , ωq.Suppose there is a vertex v P Ş tCpS n , ωq | n P Nu.By construction v is neither dominating ω nor is a vertex on R.
Moreover, note that |S n | ď |S n`1 | for all n P N, since S n`1 would have been a candidate for S n as well.In particular, there is no S n -S n`1 separator S of order less than |S n | for every n P N, since this would also have been a candidate for S n .Hence by Theorem 2.1 there is a set of |S n | many disjoint S n -S n`1 paths P n in GrS n , S n`1 s.Nu is by construction a union of a set R of rays, since the union of the paths in P n intersect the union of the paths in P m in precisely S n`1 if m " n`1 and are disjoint if m ą n `1.These rays are necessarily ω-rays, meet every separator at most once and every s P S n is contained in one of them, proving that ppS n | n P Nq, Rq is a degree witnessing pair for ω.Let k P N and let ω P ΩpGq with dompωq " 0. Then degpωq ě k if and only if for every ω-defining sequence pS n | n P Nq the sets S n eventually have size at least k.
S m and S n are disjoint for all m, n P N with n ‰ m and that CpS n`1 , ωq Ď CpS 1 f pnq`1 , ωq for all n P N. Hence pS n | n P Nq is an ω-defining sequence.Proof.Suppose degpωq ě k.Let pS n | n P Nq be any ω-defining sequence.Then each ray out of a set of k disjoint ω-rays has to go through eventually all S n .For the other direction take a degree witnessing pair ppS n | n P Nq, Rq.Now |R| ě k, since eventually all S n have size at least k.Corollary 5.5.Let k P N and let ω P ΩpGq with dompωq " 0. Then degpωq " k if and only if k is the smallest integer such that there is an ω-defining sequence pS n | n P Nq with |S n | " k for all n P N.
b) There is a subset U 2 Ď U with |U 2 | " κ such that G either contains a subdivided star or frayed star whose set of leaves is U 2 , or a subdivided frayed comb whose set of teeth is U 2 .(Notethatif cf κ is uncountable, only one of the former two can exist.)Moreover,if these statements hold, we can choose U 1 " U 2 .Even though this Frayed-Star-Comb Lemma has a much more elementary proof, we state it here only as a corollary of our main theorem.Now Theorems 6.7 and 7.19 give us the tools to prove the statement we originally wanted to prove instead of Lemma 4.4.
Corollary 8.2.Let k P N, let A Ď V pGq be infinite and k-connected in G and let κ ď |A| be an infinite cardinal.Then for every v P V pGq there is a subsetA 1 Ď A with |A 1 | " κ such that A 1 is pk ´1q-connected in G ´v.Proof.First we apply Theorem 6.7 or Theorem 7.19 to A to get a k-typical graph T and an inflated subgraph H witnessing that T is an fbs-minor of G with some A 2 Ď A along its core such that |A 2 | " κ.Let us call a vertex of T essential, if either ‚ it is a vertex of the finite side of K k,κ if T " K k,κ ;