Tilings in vertex ordered graphs

Over recent years there has been much interest in both Tur\'an and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In the (unordered) graph setting, this problem was resolved by K\"uhn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009]. We use our general framework to resolve the perfect $H$-tiling problem for all ordered graphs $H$ of interval chromatic number $2$. Already in this restricted setting the class of extremal examples is richer than in the unordered graph problem. In the process of proving our results, novel approaches to both the regularity and absorbing methods are developed.


Introduction
Over recent years there has been interest in extending classical graph theory results to the setting of vertex ordered graphs. A (vertex) ordered graph or labelled graph H on h vertices is a graph whose vertices have been labelled with [h] := {1, . . . , h}. An ordered graph G with vertex set [n] contains an ordered graph H on [h] if (i) there is a mapping φ : [h] → [n] such that φ(i) < φ(j) for all 1 ≤ i < j ≤ h and (ii) φ(i)φ(j) is an edge in G whenever ij is an edge in H.
A foundation stone in extremal graph theory is Turán's theorem which determines the number of edges in the densest K r -free graph on n vertices. Furthermore, for every graph H, the Erdős-Stone-Simonovits theorem [7,8] determines, up to a small error term, the number of edges in the densest H-free n-vertex graph. It is natural to seek Turán-type results in the setting of ordered graphs. Indeed, this question was first raised by Füredi and Hajnal [10], and there are now many results in the area; see Tardos [28] for a survey of such results (and the related problem of Turántype results for edge ordered graphs). In particular, Pach and Tardos [25] proved an analogue of the Erdős-Stone-Simonovits theorem in the setting of ordered graphs. In their result they show that the so-called interval chromatic number governs the threshold (for graphs H of interval chromatic number at least 3), rather than the chromatic number (as is the case in the unordered setting). There are several Turán-type results for ordered graphs of interval chromatic number 2; see e.g. [12,13,20,24,25,28], as well as Turán-type results for ordered hypergraphs, see [11].
There have also been a number of recent results concerning Ramsey theory for ordered graphs, for example see the work of Balko, Cibulka, Král and Kynčl [2] and of Conlon, Fox, Lee and Sudakov [5].
In this paper we initiate the study of embedding spanning structures in ordered graphs. In particular, we study the minimum degree required to ensure an ordered graph has a perfect Htiling. In both the ordered and unordered settings, an H-tiling in a graph G is a collection of vertex-disjoint copies of H contained in G. An H-tiling is perfect if it covers all the vertices of G.
Perfect H-tilings are also often referred to as H-factors, perfect H-packings or perfect H-matchings. H-tilings can be viewed as generalisations of both the notion of a matching (which corresponds to the case when H is a single edge) and the Turán problem (i.e. a copy of H in G is simply an H-tiling of size one). [22]). Let δ(H, n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-tiling. The definition of hcf(H) = 1 is somewhat involved; see [22,Section 1.2] for the definition and several illuminating examples. The moral behind the dichotomy in Theorem 1.3, however, is rather straightforward to articulate. Indeed, it arises as there are two types of extremal construction for this problem: so-called space barriers (which 'dominate' when χ * (H) = χ cr (H)) and divisibility barriers (which 'dominate' when χ * (H) = χ(H)).

Theorem 1.3 (Kühn and Osthus
In this paper we show that the corresponding problem for ordered graphs has a rich behaviour. Indeed, our main result resolves the problem for all ordered graphs H of interval chromatic number 2. Even in this restricted case the nature of the minimum degree threshold is diverse, with a range of extremal examples coming into play, including a construction which is neither a divisibility nor space barrier. Whilst we do not resolve the problem for all ordered graphs H, in Section 4 we introduce a framework that can be used to attack the problem in general. Moreover, another contribution of the paper is the approaches we develop. Indeed, as we will discuss in Section 3 we develop an approach to applying Szemerédi's regularity lemma [27] and a (local-global ) philosophy for absorbing, both of which we believe are applicable to other embedding problems for ordered graphs. In particular, a key property of the regularity method -which is regularly used to help embed (spanning) subgraphs in the unordered setting -breaks down for ordered graphs; we introduce an approach to overcome this.
1.1. Our results. Denote by δ < (H, n) the smallest integer k such that every ordered graph G whose order n is divisible by |H| and with δ(G) ≥ k contains a perfect H-tiling. The goal of this paper is to study δ < (H, n); for this we will need a few definitions. In particular, while the chromatic number is a relevant parameter in the study of perfect H-tilings in graphs, the interval chromatic number plays a role in the study of perfect H-tilings in ordered graphs. Definition 1.4 (Interval chromatic number). The interval chromatic number χ < (H) of an ordered graph H is the minimum number of intervals the vertex set [h] of H can be partitioned into, so that no two vertices belonging to the same interval are adjacent in H.
As well as χ < (H), another parameter α * (H) plays a role in the study of perfect H-tilings for ordered graphs with χ < (H) = 2. To introduce α * (H) we need the following definitions.
The following proposition shows that for any ordered graph H, the parameter α * (H) provides a lower bound for δ < (H, n). Proposition 1.5. Let H be an ordered graph on h vertices. For every n ∈ N with h|n, there is an n-vertex ordered graph G with δ(G) ≥ (1 − α * (H))n − 1 that does not contain a perfect H-tiling.
The main goal of this paper is to determine the asymptotics of δ < (H, n) for graphs H with interval chromatic number 2. It turns out the value of δ < (H, n) in this case depends on structural properties of H encapsulated by the following three definitions.  Our main theorem shows that for any ordered graph H with interval chromatic number 2, either its interval chromatic number χ < (H) or the new graph parameter α * (H) governs the minimum degree threshold that forces the existence of a perfect H-tiling in ordered graphs of large minimum degree. Theorem 1.9. Let H be an ordered graph on h vertices with χ < (H) = 2.
(i) Suppose that H does not have Property A. Then (ii) Suppose that H has both Property A and Property B. Then (iii) Suppose that H has Property A but not Property B, and one of the vertices 1, h has Property C. Then (iv) Suppose that H has Property A but not Property B, and neither of the vertices 1, h has Property C. Then In all cases of Theorem 1.9, except Case (iv), the minimum degree threshold is at least (1/2 + o(1))n. Furthermore, although the degree threshold for graphs in Cases (ii) and (iii) are the same, they have different types of extremal examples at work. Shortly in Section 2, we will show that there are three types of extremal examples. Indeed, space barriers yield Proposition 1.5 and therefore give the lower bound in Cases (i) and (iv), divisibility barriers provide the lower bound for Case (ii) and local barriers provide the lower bound in Case (iii) -while we recall that for unordered graphs there are only space and divisibility barriers (see [22] , we write X < Y if x < y for all x ∈ X and y ∈ Y . When X consists of a single element x, we simply write x < Y . A vertex is isolated if it has no neighbors. An empty graph on n vertices consists of n isolated vertices with no edges. The empty graph on 0 vertices is called the null graph. For an ordered graph G and a linearly ordered set A ⊆ V (G), the induced subgraph G[A] is the subgraph of G whose vertex set is A and whose edge set consists of all of the edges of G with both endpoints in A. We define G \ X := G[V (G) \ X]. For two disjoint subsets A, B ⊆ V (G), the induced bipartite subgraph G[A, B] is the subgraph of G whose vertex set is A ∪ B and whose edge set consists of all of the edges of G with one endpoint in A and the other endpoint in B. For convenience, we also write G[A, A] := G[A].
Given an (ordered) graph G, a vertex x ∈ V (G) and a set X ⊆ V (G), we define d G (x, X) to be the number of neighbors that x has in X.
For two ordered graphs G 1 and G 2 with disjoint vertex sets, the join graph, denoted by G 1 * G 2 , is the ordered graph obtained from G 1 and G 2 by adding all edges between V (G 1 ) and V (G 2 ), and where the vertices are ordered so that V (G 1 ) < V (G 2 ) and both V (G 1 ) and V (G 2 ) preserve their orders from G 1 and G 2 respectively. Given an unordered graph G and a positive integer t, let G(t) be the graph obtained from G by replacing every vertex x ∈ V (G) by a set V x of t vertices spanning an independent set, and joining u ∈ V x to v ∈ V y precisely when xy is an edge in G. That is we replace the edges of G by copies of K t,t . We will refer to G(t) as a blown-up copy of G.
Throughout the paper, we omit all floor and ceiling signs whenever these are not crucial. The constants in the hierarchies used to state our results are chosen from right to left. For example, if we claim that a result holds whenever 0 < a b c ≤ 1, then there are non-decreasing functions f : (0, 1] → (0, 1] and g : (0, 1] → (0, 1] such that the result holds for all 0 < a, b, c ≤ 1 with b ≤ f (c) and a ≤ g(b). Note that a b implies that we may assume in the proof that, for example, a < b or a < b 2 .
1.3. Organisation of paper. In the next section we describe the extremal examples which show that our main result is best possible. In Section 3 we give a high-level overview of our approach to the regularity and absorbing methods in the ordered setting. In Section 4, we introduce a general framework for attacking ordered tiling problems, and show how to use it to prove Theorem 1.9. In particular, in this section we state a so-called 'almost perfect tiling' theorem (Theorem 4.3) and two absorbing theorems (Theorems 4.1 and 4.6). In Section 5, we formally state Szemerédi's regularity lemma and introduce related tools. We then prove Theorem 4.3 in Section 6, and prove Theorems 4.1 and 4.6 in Section 7. We close the paper with some concluding remarks in Section 8.

Extremal examples
2.1. Space barriers. We begin this section with the proof of Proposition 1.5 which provides a general lower bound on δ < (H, n) for all ordered graphs H.
The following observation will be useful. Suppose that G 1 and H 1 are ordered graphs and G 1 and H 1 are obtained from G 1 and H 1 respectively by reversing the ordering on V (G 1 ) and V (H 1 ). Then clearly G 1 contains a perfect H 1 -tiling if and only if G 1 contains a perfect H 1 -tiling. Further, |H 1 | . Proof of Proposition 1.5. By the observation above, without loss of generality we may assume that α * = α + /( · h) for some 1 ≤ < χ < (H). Therefore to prove the proposition, it is sufficient to prove that for every 1 ≤ < χ < (H), there is an n-vertex graph G with δ(G) ≥ 1 − α + ·h n − 1 that does not contain a perfect H-tiling.
For simplicity, we set s : where G i is an empty graph defined on A i for every 1 ≤ i ≤ , and G +1 is a complete graph defined on [n] − i=1 A i . Note that n − s − 1 ≥ 0 as α + < α + χ<(H) = h. Therefore, G +1 is well-defined, and could be a null graph (only when h = n).
We claim that for every copy of H in G, If not, then there exists a copy of H in G with the vertices v 1 < v 2 < . . . < v h such that v α + +1 ∈ [s + 1]. In particular, there exists an integer k ≤ such that v α + +1 ∈ A k . By the maximality of α + , the vertex v α + +1 has a neighbor v in H, where ∈ [α + −1 + 1, α + ]. Since A k is an independent set of G, this implies that there exists an integer k −1 < k such that v α + −1 +1 ∈ A k −1 . Repeat this process until we reach to A 1 . Then we obtain an integer 1 ≤ 0 ≤ and a sequence of numbers However, we run out of the space for v 0 as A 1 is an independent set with the smallest vertices.
Finally, suppose that G has a perfect H-tiling H. Then by (3) we have which contradicts the definition of a perfect H-tiling.
We refer to such examples G as space barriers as, in this case, the obstruction to G containing a perfect H-tiling is that the vertex class [s + 1] is 'too big'.

Divisibility barriers.
Proposition 2.1. Let H be an ordered graph on h vertices with χ < (H) = 2. Suppose that H has Property B. Then for every n ∈ N with h|n, there is an n-vertex ordered graph G with δ(G) ≥ n/2 − 2 that does not contain a perfect H-tiling.
Proof. Let k be the largest integer such that k ≤ n/2 and k is not divisible by h. Let G be the disjoint union of two complete graphs on vertex sets [k] and [k + 1, n]. Note that k ≥ n/2 − 1; and the minimum degree of G is min{k − 1, n − k − 1} ≥ n/2 − 2.
Suppose that G has a perfect H-tiling. Then there must be at least one copy H of H, for which both [k] ∩ V (H ) and [k + 1, n] ∩ V (H ) are non-empty. However, this is not possible for H with Property B, as there are no edges between [k] and [k + 1, n] in G.
Note we call such graphs G divisibility barriers as the obstruction to containing a perfect H-tiling is a divisibility issue (in this case, the size of each of the two cliques is not divisible by h).

Local barriers.
Proposition 2.2. Let H be an ordered graph on h vertices with χ < (H) = 2. Suppose that one of the vertices 1, h has Property C. Then for every n ∈ N with h|n, there is an n-vertex ordered graph G with δ(G) = n/2 that does not contain a perfect H-tiling.
Proof. Without loss of generality, we assume that the vertex h has Property C. Recall that s(H) is the smallest vertex in H that is adjacent to h. Since h has Property C, there exists an edge ab in H such that Let G := G 1 * G 2 , where G 1 , G 2 are empty graphs on the vertex sets [1, n/2 −1] and [ n/2 , n− 1]. Then we construct an ordered graph G from G by adding the vertex n and all edges between n and [ n/2 , n − 1].
Suppose that G has a perfect H-tiling. Then there must be a copy of H in G such that n plays the role of h in it. By the construction of G, the image of s(H) in G lies in [ n/2 , n − 1]. Then by (4), the images of a, b in G must lie in [ n/2 , n − 1]. This contradicts the fact that [ n/2 , n − 1] is an independent set.
We call such graphs G local barriers as the reason G does not contain a perfect H-tiling is a localized issue (in this case, there is a vertex that does not lie in a single copy of H).

3.
Applying the regularity and absorbing methods in the ordered setting 3.1. The regularity method. In this subsection we explain our approach to applying the regularity lemma in the ordered graph setting. Those readers unfamiliar with this result and related concepts should first read Section 5.
Let A 1 , . . . , A k be large disjoint equal size vertex classes in an (unordered) graph G so that each pair (A i , A j ) (for distinct i, j ∈ [k]) is ε-regular of density at least d, where 0 < ε < d. Such a structure is often found in an application of Szemerédi's regularity lemma and provides a framework for embedding subgraphs H with χ(H) = k into G. Indeed, it is well-known that such a structure contains all fixed size subgraphs H of chromatic number at most k (see Lemma 5.7). In fact, for any fixed subgraph H with χ(H) = k, G[A 1 ∪ · · · ∪ A k ] must contain an almost perfect H-tiling. Moreover, the famous blow-up lemma of Komlós, Sárközy and Szemerédi [17] allows one to embed any almost spanning, bounded degree graph F with χ(F ) = k into G[A 1 ∪· · ·∪A k ]. These properties have been used in dozens of applications of the regularity lemma.
Ideally one would like to use such properties in the vertex ordered setting. Similarly as before, let A 1 , . . . , A k be large disjoint equal size vertex classes in an n-vertex ordered graph G so that each pair (A i , A j ) (for distinct i, j ∈ [k]) is ε-regular of density at least d. Thus now the A i s are subsets of [n]. Let H be a fixed ordered graph with χ < (H) = k. One can find a copy of H in G[A 1 ∪ · · · ∪ A k ]: as demonstrated by Lemma 6.2 in Section 6, one can find large subclasses S i ⊆ A i for all i ∈ [k] and a permutation σ of [k] so that S σ(1) < S σ(2) < . . . < S σ(k) . This allows us then However, in general it is far from true that G[A 1 ∪ · · · ∪ A k ] should contain an almost perfect H-tiling. To see this consider the case when k = 2 and H is the ordered path 213. Suppose

the vertex playing the role of 1) and two vertices in
can only cover at most half of A 1 .
At first sight this suggests perhaps the regularity method is not suitable for embedding large structures in ordered graphs. However, in this paper we demonstrate a method for overcoming this difficulty. Suppose we wish to embed an (almost) perfect H-tiling in an ordered graph G where is ε-regular of density at least d; now (i) k may be significantly bigger than r and (ii) the size of the classes A i may be far from equal. The class sizes and k are chosen so that however the vertices from As we now explain, with this property to hand, one can easily find an almost perfect H-tiling in G[A 1 ∪ · · · ∪ A k ]. Indeed, delete the vertices from H. Still each pair (A i , A j ) is 2ε-regular and the ratios of the classes have been preserved. So we can find a small H-tiling in G[A 1 ∪ · · · ∪ A k ] as before. Repeating this process allows us to cover almost all the vertices in The challenge is to choose k not too large (else G will not be dense enough to guarantee such an ε-regular structure G[A 1 ∪ · · · ∪ A k ]) whilst ensuring the chosen ratios |A i |/|A j | have the 'ratio preservation' property described above. This latter point motivates the notion of a bottlegraph of H introduced in the next section.
3.2. The absorbing method. The so-called absorbing method, pioneered by Rödl, Ruciński and Szemerédi (see e.g. [26]) has proved an immensely powerful technique for embedding problems in graphs and hypergraphs. In particular, when one wishes to embed a spanning structure F in a (hyper)graph G, the method can provide a certain 'absorbing gadget' Abs in G. With this gadget to hand, one then seeks to embed only an almost spanning subgraph F of F into G; Abs will then have the power to extend the subgraph F into a copy of F in G. If the structure F we seek is a perfect H-tiling, then we will say Abs is an H-absorbing set.
The now standard approach to construct H-absorbing sets for perfect H-tilings in (hyper)graphs originates from a paper of Lo and Markström [23]. Indeed, suppose one wishes to find a perfect H-tiling in an n-vertex graph G where h := |H|. In the simplest case, they show that to construct an H-absorbing set in G it suffices to show that for every pair For many problems this property is relatively easy to establish. For example, in the (unordered) graph setting, a simple application of the regularity lemma can be used to establish this property when G is an n-vertex graph with minimum degree more than (1 − 1/χ(H) + o(1))n.
However, the analogous statement for ordered graphs is in general far from true. In particular, there are ordered graphs H and G where δ(G) is much greater than (1 − 1/χ < (H))n and yet there exist pairs of vertices x, y ∈ V (G) for which no (h − 1)-set is good for x, y. For example, for any n ∈ N divisible by 3, consider the complete 3-partite ordered graph G on [n − 2] with classes A 1 , A 2 , A 3 of sizes n/3 − 1, n/3 − 1 and n/3 respectively, and where A 1 < A 2 < A 3 . Obtain G from G by adding vertices n − 1 and n where n is adjacent to every vertex in A 1 ∪ A 3 and n − 1 is adjacent to every vertex in A 2 ∪ A 3 . Now choose H to be the ordered graph obtained from the complete bipartite graph K 2,2 by labelling the elements in the first vertex class 1, 2; the elements in the second class 3, 4. Observe that χ < (H) = 2. Notice that δ(G) = 2n/3 − 1 and yet there are no good 3-sets X for n − 1, n. Indeed, this follows because any copy of H containing n in G \ {n − 1} must use vertices in A 1 to play the role of 1, 2, whilst any copy of H containing n − 1 in G \ {n} must use vertices in A 2 to play the role of 1, 2.
Despite this difficultly, in Theorem 4.1, we are able to show the existence of an H-absorbing set in any n-vertex ordered graph G with δ(G) > (1 − 1/χ < (H) + o(1))n (for every fixed ordered graph H). The key is that, as made precise by Lo and Markström [23], to obtain an H-absorbing set in G it is also sufficient to prove that for any x, y ∈ V (G) there are 'many' good sets X ⊆ V (G) of the same fixed (constant) size so that both G[X ∪ {x}] and G[X ∪ {y}] contain perfect H-tilings; see Lemma 7.1 below.
The way we construct such good sets X for every x, y ∈ V (G) can be summarized by the following process -a philosophy to absorbing that we term local-global absorbing.
• Step 1: local absorbing. Prove that for any x ∈ V (G), most y 'close' to x (with respect to the ordering on V (G) = [n]) are such that there are many good (h − 1)-sets for x, y. • Step 2: global absorbing. Piece together chains of the 'local' good sets found in Step 1 to prove that for any pair x, y ∈ V (G) there are many good sets X of bounded size for x, y.
Note that our illustrative example above shows in general one cannot hope to replace the word most in Step 1 with all. On the other hand, the intuition why it is often easier to find good sets for x, y ∈ V (G) where x and y have labels close together is that such x and y can often play the role of the same vertex in copies of H. The above two-step process is sufficient to prove Theorem 4.1; Step 1 corresponds to Lemma 7.2 and Step 2 to Lemma 7.3.
On the other hand, to prove (a general case of) the other of our absorbing theorems (Theorem 4.6) we need a variant of this approach. there are many good (h − 1)-sets for x, y. • Step 3: global absorbing. Piece together chains of our good sets found in Steps 1 and 2 to prove that for any pair x, y ∈ V (G) there are many good sets X of bounded size for x, y.

The intuition why in
Step 2 it is often easier to find good sets for x, y ∈ V (G) where x is close to 1 and y is close to n is as follows: in 'most' copies of H in G containing x, x must play the role of 1, whist in 'most' copies of H in G containing y, y must play the role of h.

4.
A general framework and the proof of Theorem 1.9 4.1. General framework. In this section we introduce two theorems, which as well as being tools in the proof of our main result, are applicable to the general perfect H-tiling problem for ordered graphs.
First, as described in the previous section, we adapt the absorbing method to the setting of ordered graphs. Let H be an ordered graph. Given an ordered graph G, a set S ⊆ V (G) is an Habsorbing set for Q ⊆ V (G), if both G[S] and G[S ∪Q] contain perfect H-tilings. In this case we say that Q is H-absorbed by S. Sometimes we will simply refer to a set S ⊆ V (G) as an H-absorbing set if there exists a non-empty set Q ⊆ V (G) \ S that is H-absorbed by S. Roughly speaking, the following result provides an absorbing set Abs in an ordered graph G of large minimum degree, where crucially Abs is an H-absorbing set for every not too large set of vertices Q ⊆ V (G) \ Abs.
Theorem 4.1 (Absorbing theorem). Let H be an h-vertex ordered graph and let η > 0. Then there exists an n 0 ∈ N and 0 < ν η so that the following holds. Suppose that G is an n-vertex ordered graph where n ≥ n 0 and where Then V (G) = [n] contains a set Abs so that • |Abs| ≤ νn; • Abs is an H-absorbing set for every W ⊆ V (G) \ Abs such that |W | ∈ hN and |W | ≤ ν 3 n.
Theorem 4.1 suffices for our applications in most cases. Indeed, it is immediately applicable to the perfect H-tiling problem for any ordered graph H where the minimum degree threshold for ensuring a perfect H-tiling in an ordered graph G is at least (1 − 1 χ<(H) + o(1))|G|. In particular, we will use this theorem for Cases (i)-(iii) of Theorem 1.9. For Case (iv) (and we suspect at least for some special cases of the general perfect H-tiling problem) we require an absorbing theorem for ordered graphs with much smaller minimum degree. In this situation, some structural properties of H can help us improve the absorbing argument; see Theorem 4.6 below.
As indicated above, to apply the absorbing method one requires a sister almost perfect tiling theorem, which usually states that in a graph with large minimum degree all but o(n) vertices are covered by some H-tiling. Although the variety of extremal examples indicates that proving a sharp almost perfect H-tiling theorem for an arbitrary ordered graph H seems to be very difficult, in this section we propose a general framework for obtaining such almost perfect tiling theorems.
Let B be a complete k-partite unordered graph with parts U 1 , . . . , U k , and σ be a permutation of the set [k]. An interval labeling of B with respect to σ is a bijection φ : where the V x s are sets of t independent vertices. Let (B(t), φ) be the ordered graph obtained from B(t) by equipping V (B(t)) with a vertex ordering, satisfying V x < V y for every x, y ∈ V (B) with φ(x) < φ(y). We refer to (B(t), φ) as an ordered blow-up of B.  With Theorem 4.3 at hand, in order to prove that all ordered graphs with a given minimum degree contain an almost perfect H-tiling, it is sufficient to show that certain 'interval labelled' blow-ups of a specific graph B with a given critical chromatic number contain perfect H-tilings. The latter statement usually can be verified easily by observation or by solving a linear optimization problem. Thus, for the general perfect H-tiling problem, the heart of the problem is to choose an ordered graph B whose critical chromatic number is not too big (so that the corresponding minimum degree condition in Theorem 4.3 is not too high) whilst ensuring B is indeed a bottlegraph assigned to H. These competing forces mean it is far from immediate what the correct choice of B is given an arbitrary ordered graph H. However, for a given class of ordered graphs H (as we will see in the case when χ < (H) = 2), there might be some intuitive ways to construct a 'fairly good' or even optimal bottlegraph using their structural properties.
The proofs of both Theorems 4.1 and 4.3 rely on Szemerédi's regularity lemma, which will be formally introduced in Section 5. We then prove Theorem 4.3 in Section 6, and Theorem 4.1 in Section 7.

4.2.
Graphs with interval chromatic number 2. In this section, we illustrate how to apply our general framework to prove Theorem 1.9. The following key lemma gives a construction of the bottlegraph for graphs with interval chromatic number 2.
Proof. Let a, r be the integers such that h = ap + r, where a ≥ 2 and 0 ≤ r < p. Then we define B to be the complete multipartite graph with classes U 0 , U 1 , . . . , U a , in which |U 0 | = r, and |U 1 | = . . . = |U a | = p. We will show that B is a bottlegraph assigned to H. Let φ be an interval labeling of B. If r = 0, then by (5), (B, φ) immediately contains a copy of H (i.e. (B(1), φ) contains a perfect H-tiling). So assume r = 0. If there exists i ≥ 1 such that φ(U i ) < φ(U 0 ), then again (B, φ) contains a copy of H. Therefore, without loss of generality, we can assume that φ(U 0 ) < φ(U 1 ) < . . . < φ(U a ).
Let c := lcm(p, r), the least common multiple of p and r, and t := c/r. Let B := B(t) and U 0 , U 1 , . . . , U a be the partite sets of B (where U i corresponds to U i ). For a set A ⊆ V (B ) of size h, if |A ∩ U i | = p, for all 0 ≤ i ≤ a − 1 and |A ∩ U a | = r, we say A is a type I set; if |V (H) ∩ U 0 | = 0, |V (H) ∩ U i | = p, for all 1 ≤ i ≤ a − 1 and |V (H) ∩ U a | = p + r, we say A is a type II set. Both type I and type II sets induce some complete multipartite graphs in B , which contain a copy of H by (5). By the choice of c and t, V (B ) can be partitioned into t = c/r disjoint sets, where c/p of them are of type I and c/r − c/p of them are of type II; thus this ensures a perfect H-tiling in B . So indeed B is a bottlegraph of H. Moreover, it is easy to compute that  For every η > 0, there exists an n 0 ∈ N so that the following holds. Every ordered graph G on n ≥ n 0 vertices with contains an H-tiling covering all but at most ηn vertices.
Proof of Theorem 1.9(i)-(iii). Our desired lower bounds on δ < (H, n) follow immediately from the extremal examples in Section 2. More specifically, the lower bound in (i) is given by the space barriers, i.e. Proposition 1.5; the lower bound in (ii) is given by the divisibility barriers, i.e. Proposition 2.1; the lower bound in (iii) is given by the local barriers, i.e. Proposition 2.2.
For an arbitrary small constant 0 < η < 1, let ν be defined as in Theorem 4.1, and fix an additional constant η satisfying the following: Let n be a sufficiently large integer divisible by h. The proof of Theorem 1.9(iv) is similar but requires a stronger version of the absorbing theorem.
Theorem 4.6. Let H be an h-vertex ordered graph with χ < (H) = 2. Suppose that H has Property A but not Property B, and neither of the vertices 1, h has Property C. Then for every η > 0, there exists an n 0 ∈ N and ν > 0 so that the following holds. Suppose that G is an n-vertex ordered graph where n ≥ n 0 and where δ(G) ≥ ηn. Then V (G) contains a set Abs so that • |Abs| ≤ νn; • Abs is an H-absorbing set for any W ⊆ V (G) \ Abs such that |W | ∈ hN and |W | ≤ ν 3 n.
The proof of Theorem 4.6 contains some technical arguments; we postpone it to Section 7.
Proof of Theorem 1.9(iv). The upper bound on δ < (H, n) follows similarly from Theorems 4.5 and 4.6, while the lower bound is given by the space barriers, i.e. Proposition 1.5.

The regularity lemma and related tools
In the proof of our main results we will use Szemerédi's regularity lemma [27]. In this section we will introduce all the information we require about this result. We first introduce some notation. The density of a bipartite graph with vertex classes A and B is defined to be We now collect together some useful properties of ε-regular pairs.
Proposition 5.1. For 0 < ε d 2 < d 1 ≤ 1, there exists an integer K = K(ε, d 2 , d 1 ) such that the following holds. Let (A, B) G be an ε-regular pair of density d 1 in a graph G where |A|, |B| ≥ K. Then there exists a spanning subgraph G ⊆ G such that (A, B) G is a √ ε-regular pair of density d, where |d − d 2 | ≤ ε.
Proof sketch. It suffices to consider the case when d 1 − d 2 ≥ ε (otherwise we set G := G). Let G be the graph obtained from G by retaining each edge with probability p := d 2 /d 1 , independently of all other edges. Then E(d G (A, B) Further, for every X ⊆ A and Y ⊆ B such that |X| ≥ ε|A| and |Y | ≥ ε|B|, we have that Noting that there are at most 2 |A|+|B| such pairs X, Y , we may repeatedly apply Chernoff's bound to ensure with high probability the conclusion of the proposition holds. (A, B 2 ) G be ε-regular pairs of density d 1 and d 2 respectively in a graph G where B 1 and B 2 are disjoint. Then If both |Y 1 | ≥ ε|B 1 | and |Y 2 | ≥ ε|B 2 | then it is easy to check the pair X, Y satisfies the condition in the definition of a √ ε-regular pair. So without loss of generality it suffices to check the case when |Y 1 | ≥ ε|B 1 | and |Y 2 | ≤ ε|B 2 |. In this case |Y 2 |/|Y | ≤ √ ε. Thus, This proves that the pair X, Y satisfies the condition in the definition of a √ ε-regular pair.
We will also make use of the following well-known property of regular pairs (see e.g., [19, Fact 1.5]). We will apply the following degree form of Szemerédi's regularity lemma [27].
Lemma 5.4 (Regularity lemma). For every ε > 0 and 0 ∈ N there exists L 0 = L 0 (ε, 0 ) such that for every d ∈ [0, 1] and for every graph G on n ≥ L 0 vertices there exists a partition V 0 , V 1 , . . . , V of V (G) and a spanning subgraph G of G, such that the following conditions hold: We call V 1 , . . . , V clusters, V 0 the exceptional set and the vertices in V 0 exceptional vertices. We refer to G as the pure graph. The reduced graph R of G with parameters ε, d and 0 is the graph whose vertices are V 1 , . . . , V and in which V i V j is an edge precisely when (V i , V j ) G is (ε, d)-regular.
A t-partite graph with parts W 1 , . . . , W t is nearly balanced if ||W i | − |W j || ≤ 1 for every 1 ≤ i, j ≤ t. We will also make use of the following multipartite version of Lemma 5.4. Lemma 5.5 (Multipartite regularity lemma). Given any integer t ≥ 2, any ε > 0 and any 0 ∈ N there exists L 0 = L 0 (ε, t, 0 ) ∈ N such that for every d ∈ [0, 1] and for every nearly balanced t-partite graph G = (W 1 , . . . , W t ) on n ≥ L 0 vertices, there exists an ∈ N, a partition W 0 i , W 1 i , . . . , W i of W i for each i ∈ [t] and a spanning subgraph G of G, such that the following conditions hold: Similarly as before, for i ∈ [t] and j ∈ [ ] we call the W j i clusters, the W 0 i the exceptional sets and the vertices in the W 0 i exceptional vertices. We refer to G as the pure graph. The reduced graph R of G with parameters ε, d and 0 is the graph whose vertices are the W j i (where i ∈ [t] and j ∈ [ ]) and in which W j i W j i is an edge precisely when (W j i , W j i ) G is (ε, d)-regular. The following well-known corollary of the regularity lemma shows that the reduced graph almost inherits the minimum degree of the original graph.
Proposition 5.6. Let 0 < ε, d, k < 1, G be an n-vertex graph with δ(G) ≥ kn and R be the reduced graph of G obtained by applying the regularity lemma with parameters ε, d.
The next key lemma allows us to use the reduced graph R of G as a framework for embedding subgraphs into G.
Lemma 5.7 (Key lemma [19]). Suppose that 0 < ε < d, that q, t ∈ N and that R is a graph with V (R) = {v 1 , . . . , v k }. We construct a graph G as follows: replace every vertex v i ∈ V (R) with a set V i of q vertices and replace each edge of R with an (ε, d)-regular pair. For each v i ∈ V (R), let U i denote the set of t vertices in R(t) corresponding to v i . Let H be a subgraph of R(t) with maximum degree ∆ and set h := |H|. Set δ := d − ε and ε 0 := δ ∆ /(2 + ∆). If ε ≤ ε 0 and t − 1 ≤ ε 0 q then there are at least Our applications of Lemma 5.7 will take the following form: suppose within an ordered graph G we have vertex classes V 1 < . . . < V k so that each pair (V i , V j ) G is (ε, d)-regular. Then Lemma 5.7 tells us G contains (many) copies of any fixed size ordered graph H with χ < (H) = k, where the ith vertex class of H is embedded into V i .

Proof of Theorem 4.3
We will apply the following result of Komlós [16]; this result shows that the critical chromatic number of H governs the minimum degree threshold for the existence of almost perfect H-tilings in unordered graphs. Theorem 6.1 (Komlós [16,Theorem 8]). Let µ > 0 and let F be an unordered graph. Then there exists an n 0 = n 0 (µ, F ) ∈ N such that every graph G on n ≥ n 0 vertices with contains an F -tiling covering all but at most µn vertices.
The next result ensures that in any k linear size disjoint vertex sets A 1 , . . . , A k of an ordered graph G, one can find 'nicely ordered' linear size subsets S i of each A i . As we will see shortly, this property is crucial for our application of the regularity lemma in the proof of Theorem 4.3. As pointed out by a referee, it is also a special case of the 'same type lemma' of Bárány and Valtr [4]. Lemma 6.2. For n ≥ k ≥ 2, let A 1 , A 2 , . . . , A k be nonempty disjoint subsets of [n]. Then there exist sets S 1 , S 2 , . . . , S k , where S i ⊆ A i , and a permutation σ = (σ(1), σ(2), . . . , σ(k)) of the set [k], such that the following conditions hold for all i, j ∈ [k]: Proof. By removing elements if necessary we may assume that each A i contains a multiple of k elements. Given any i ∈ [k], we refer to the jth smallest number in A i as the jth element of A i . Let

Continuing in this way we define sets
. This immediately implies the lemma.
Proof of Theorem 4.3. We will fix additional constants satisfying the following hierarchy Moreover, we choose an integer 0 such that 0 ≥ n 0 (µ 1 , B), where n 0 (µ 1 , B) is as defined in Theorem 6.1. In what follows, we assume that the order n of our given ordered graph G is sufficiently large for our estimates to hold. We now apply the regularity lemma (Lemma 5.4) with parameters ε 1 , d and 0 to G to obtain a reduced graph R, clusters {V a , a ∈ V (R)}, an exceptional set V 0 , and a spanning subgraph G ⊆ G. Inequality (7) together with Proposition 5.6 implies that Since |R| ≥ 0 ≥ n 0 (µ 1 , B), we can apply Theorem 6.1 to R to find a B-tiling B covering all but at most µ 1 |R| vertices. We delete all the clusters not contained in some copy of B in B from R and add all the vertices lying in these clusters to the exceptional set V 0 . Thus, |V 0 | ≤ ε 1 n + µ 1 n ≤ 2µ 1 n. From now on, we denote by R the subgraph of the reduced graph induced by all the remaining clusters. Thus B now is a perfect B-tiling of R.
Fix an arbitrary copy B ∈ B with partite sets U 1 , . . . , U k , and let A := a∈V (B) V a and A i := a∈U i V a . Since B is a complete multipartite graph, repeatedly applying Propositions 5.1 and 5.2 to G [A], we can find a spanning subgraph G ⊆ G [A] such that for every distinct i, j ∈ [k], The idea is that G is a blow-up of the bottlegraph B, where the complete bipartite graphs between vertex classes are replaced by (ε 2 , d − ε 2 )-regular pairs. We now show that this bottlegraph-like structure will ensure that G contains an almost perfect H-tiling. Then repeating this process for every B ∈ B will ensure the desired almost perfect H-tiling in G.
Let α := 1/(2k). By Lemma 6.2, there exist sets S i ⊆ A i , and a permutation σ of [k] such that |S i | ≥ α|A i |, and S i < S j whenever σ(i) < σ(j). Moreover, by the slicing lemma (Lemma 5.3), we have that each (S i , S j ) G is (ε, d − ε)-regular. Now we apply the key lemma (Lemma 5.7) on G [∪S i ], and find a blown-up copy B 1 (t) of B, where |B 1 (t) ∩ S i | = |U i |t for every i and t is a fixed integer given by the definition of the bottlegraph. Note that by the choice of the S i , B 1 naturally has an interval ordering with respect to the permutation σ, and therefore B 1 (t) has a perfect Htiling. After that, we can delete V (B 1 (t)) from A (and therefore from each A i ); crucially after this deletion, the ratio |A i |/|A j | amongst all pairs of classes A i , A j remains the same as before. Further, These properties allow us to repeatedly apply this argument, thereby obtaining an H-tiling in G[A] covering all but at most µ 2 |A| vertices. More precisely, suppose we have subsets Then by the slicing lemma, and as 2ε 2 /µ 2 √ ε 2 and ε 2 d, we have that (A i , A j ) G is an ( √ ε 2 , d/2)-regular pair for all distinct i, j ∈ [k]. Thus, we can repeatedly apply the argument in the paragraph above (now to the A i rather than the A i ), whilst still retaining property (ii) and terminating the process when we obtain subsets A i that no longer satisfy property (i). Notice that by (ii), as soon as (i) is no longer satisfied for some i ∈ [k], in fact |A i | < µ 2 |A i | for all i ∈ [k]. Thus, this process will result in an H-tiling in G[A] covering all but at most µ 2 |A| vertices.
Finally, simply repeat this process for all copies of B in B; we obtain an H-tiling of G covering all but at most (2µ 1 + µ 2 )n ≤ ηn vertices.

Proof of the absorbing theorems
To prove Theorems 4.1 and 4.6, we make use of the following, now standard, lemma.
Lemma 7.1. Let h, s ∈ N and ξ > 0. Suppose that H is an ordered hypergraph on h vertices. Then there exists an n 0 ∈ N such that the following holds. Suppose that G is an ordered hypergraph on n ≥ n 0 vertices so that, for any x, y ∈ V (G), there are at least ξn sh−1 (sh − 1)-sets X ⊆ V (G) First of all, clearly there is a partition W 1 , . . . , W t of [n] where (i) t := 8/η ; Note that (iii) implies that each of the W i s is an interval in [n].
Now let R * 1 be the induced subgraph of R 1 obtained by deleting all W j i * +1 . Thus, we have deleted precisely a (1/t)th proportion of V (R 1 ) to obtain R * 1 . Therefore, (10) and (i) imply that (11) δ . The minimum degree condition on G 1 ensures that Fix an arbitrary cluster W j * i * for some j * ∈ [ ]. Combining (11) and (12) ensures we can greedily choose clusters W j 1 i 1 , . . . , W j r−1 i r−1 so that: In particular, (c) is ensured by the choice of R * 1 and (iv). By the slicing lemma (Lemma 5.3) and the fact that By the definition of (ε 1/2 , d/2)-regularity, all but at most for each k ∈ [r − 1], and note that |W k | ≥ dη|W j k i k |/12. Given any i = j ∈ [r − 1], Lemma 5.3 implies that each pair (W i , W j ) G 1 and (W i , W j * i * ) G 1 are (ε 1/4 , d/4)-regular. Recalling that χ < (H) = r, property (c) above together with Lemma 5.7 implies that there are at least ρn h−1 (h − 1)-sets X ⊆ W j * i * ∪ W k such that both G[X ∪ {x}] and G[X ∪ {y}] span copies of H.
For each choice of the cluster W j * i * , there were at most rε 1/2 |W j * i * | 'bad' selections for y ∈ W j * i * . Since |W 0 i * | ≤ εn/t this implies that for all but at most (rε 1/2 + ε)|W i * | vertices y ∈ W i * , there are at least ρn h−1 (h − 1)-sets X ⊆ V (G) such that both G[X ∪ {x}] and G[X ∪ {y}] span copies of H.
One can argue analogously (now considering the induced subgraph R * * 1 of R 1 obtained by deleting all W j i * ) to conclude the following: for all but at most (rε 1/2 + ε)|W i * +1 | vertices y ∈ W i * +1 , there are at least ρn h−1 (h − 1)-sets X ⊆ V (G) such that both G[X ∪ {x}] and G[X ∪ {y}] span copies of H.
With Lemma 7.2 to hand, we can now prove the following result. Note that Lemma 7.3 together with Lemma 7.1 immediately imply Theorem 4.1. Indeed, applying Lemma 7.3 ensures the hypothesis of Lemma 7.1 holds, and then the latter result yields the desired absorbing set Abs. Proof. Choose ξ so that 0 < ξ ρ ρ ρ γ η where ρ and γ are as in Lemma 7.2. Let G be as in the statement of the lemma.
The idea for the proof is straightforward: we first prove the result for every x, y very close together except that instead of having s = 32/η we have s = 1 (call this Step 1). Then for x, y slightly further apart, we have many choices of z 'in the middle' of x and y. Then applying Step 1 to both x, z and y, z (and 'gluing' the structures between x and z, and z and y together) we conclude that the lemma holds for such x, y except that now s = 2. Repeating this process we deduce that for x and y of increasing distance, one can conclude that the lemma holds for such x, y, but at the expense of increasing s. From this it is easy to deduce that the lemma holds for all x, y ∈ [n] with s := 32/η . More generally, for any x, y ∈ [n], by repeated iterations of the above argument we obtain some t ≤ s such that there are at least ξ 1/2 n th−1 (th − 1)-sets X ⊆ V (G) such that both G[X ∪ {x}] and G[X ∪ {y}] contain perfect H-tilings. For each such set X we have that G \ X contains more than ρn h /2 copies of H. Add s − t such disjoint copies of H to obtain from X a set X. Then X is as desired and there are at least ξn sh−1 choices for X.  We also write a := |A|, b := |B| and c := |C|, then h = a + b + c + 3. Note that A and B ∪ C are independent sets of H, and every edge of H is either between intervals A and C, or between B and h.
Proof. Let H 1 be a complete bipartite ordered graph with parts S 1 < L 1 , where |S 1 | = a + b + 1 and |L 1 | = c + 1. For a copy of (14) and the pigeonhole principle, there exists i ∈ [3] such that Notice that for any choice of i ∈  A standard application of the regularity method shows that there exists an (ε , d )-regular pair (P , Q ) in G, where P ⊆ N ; Q ⊆ M \ N ; |P |, |Q | ≥ 2ξ 3 n. By Lemma 6.2 and Lemma 5.3, there exist sets P ⊆ P and Q ⊆ Q such that: |P |, |Q| ≥ ξ 3 n; (P, Q) is an (ε, d)-regular pair in G; either P < Q or Q < P . Case 2: Q < P . Let F 1 be the complete bipartite ordered graph with parts S 1 < L 1 , where |S 1 | = a + b + 1 and |L 1 | = c+2. Let F 2 be a complete bipartite ordered graph with parts S 2 < L 2 , where |S 2 | = a+b+2 and |L 2 | = c + 1. Note that both F 1 and F 2 contain a spanning copy of H. Let F 3 be the complete bipartite ordered graph with parts S 3 < L 3 , where |S 3 | = a + b and |L 3 | = c + 2. We say a copy of F 3 is good for x if x < S 3 < L 3 . Note that G[V (F 3 ) ∪ {x}] contains a spanning copy of H, if F 3 is good for x. Lastly, let F 4 be the complete bipartite ordered graph with parts S 4 < L 4 , where |S 4 | = a + 1 and |L 4 | = b + c + 1. We say a copy of F 4 is good for y if S 4 < L 4 < y and L 4 ⊆ N (y). Observe that G[V (F 4 ) ∪ {y}] contains a spanning copy of H, if F 4 is good for y.
Let H 3 be the complete bipartite ordered graph with parts S 3 < L 3 , where |S 3 | = b(a + b + 1) − 1 and L 3 = b(c + 2). By Lemma 5.7, there are at least ξ 2 n bh−1 copies of H 3 in G[P, Q], and for every such H 3 , we have x < S 3 < L 3 < y and L 3 ⊆ P ⊆ N (y). Note that such H 3 can be decomposed into b − 1 copies of F 1 and one good copy of For every x, y ≤ (1 − η/3)n or x ≤ ηn/3 and y ≥ (1 − η/3)n, simply adding enough disjoint copies of H to the sets obtained from Claims 7.6 or 7.7 completes the proof. For every x ≥ ηn/3 and y ≥ (1 − η/3)n, there are at least ηn/3 vertices z (i.e. the vertices in [ηn/3]) such that: (i) {y, z} satisfies the condition of Claim 7.7; (ii) {x, z} either satisfies the condition of Claim 7.6 or Claim 7.7. Applying Claims 7.6 and 7.7 on pairs {x, z} and {y, z} produces many disjoint copies of X x,z 1 ∪X y,z 2 (or X x,z 2 ∪X y,z 2 ), where X x,z 1 refers to (h−1)-sets obtained from Claim 7.6 for {x, z}, and similarly for X x,z 2 and X y,z 2 . Finally, adding enough extra disjoint copies of H to z ∪ X x,z 1 ∪ X y,z 2 (or z ∪ X x,z 2 ∪ X y,z 2 ), we show that for every x, y ∈ [n], there are at least ξn 2bh−1 = ξn sh−1 (sh − 1)-sets X ⊆ V (G) such that both G[X ∪ {x}] and G[X ∪ {y}] contain perfect H-tilings.
Proof of Theorem 4.6. Since H has property A, it satisfies the following conditions: • all edges of H are between the intervals [1, h/2 − 1] and [ h/2 + 2, h].
• if h is even, then the vertices h/2, h/2 + 1 are isolated; if h is odd, then the vertex (h + 1)/2 is isolated.
Furthermore, since H does not have property B, at least one of 1 and h must be isolated in H. We first assume that both 1, h are isolated, then by Definition 1.8 neither of them has Property C. Note that H satisfies the assumptions in Lemma 7.4. Together with Lemma 7.1 (applied with s = 1), this immediately implies Theorem 4.6. Now without loss of generality, we assume that 1 is isolated in H but h is not. Since h does not have Property C, by Definition 1.8, [s(H), h − 1] is an independent set in H. Similarly, H satisfies the assumptions in Lemma 7.5, and this, together with Lemma 7.1, completes the proof.

Concluding remarks
In this paper we have introduced a general framework for the perfect H-tiling problem in ordered graphs. This approach can be summarized as follows: Step 1: Find a candidate extremal example; an n-vertex ordered graph G with minimum degree δ(G) = αn − O(1) without a perfect H-tiling.
Despite introducing this framework, we suspect determining the perfect H-tiling threshold for an arbitrary H will be challenging in the sense that there could be a range of different extremal examples and optimal bottlegraphs, depending on the precise structure of H.
On the other hand, in the case when H is an h-vertex ordered graph and 12 ∈ E(H) or (h−1)h ∈ E(H), it is actually straightforward to deduce from Theorem 1.1 the minimum degree threshold for forcing a perfect H-tiling.
Proposition 8.1. Let n, h ∈ N such that h|n. Suppose H is an h-vertex ordered graph. If G is an n-vertex ordered graph with δ(G) ≥ (1 − 1/h)n, then G contains a perfect H-tiling.
Proof. Consider the unordered underlying graph G of any ordered n-vertex graph G with δ(G) ≥ (1 − 1/h)n. Theorem 1.1 implies that G contains a perfect K h -tiling. Since any ordered copy of K h contains H, this ensures G contains a perfect H-tiling.
For the moreover part, notice such H satisfy α * (H) = 1/h. The result then follows directly from Proposition 1.5.
In [16], Komlós determined the minimum degree threshold for an (unordered) graph to contain an H-tiling covering a given proportion of the vertices; it would be interesting to obtain an ordered analogue of this result.
Question 8.2. Let s ∈ (0, 1) and H be an ordered graph. What is the minimum degree threshold that ensures an ordered graph G contains an H-tiling covering at least an sth proportion of its vertices?
There has also been interest in Ramsey and Turán properties of edge ordered graphs (see e.g. [3,28]; it would be interesting to study the perfect H-tiling problem in this setting also. Other than tiling problems, there are many natural embedding problems to consider for ordered graphs. We now raise a couple of such problems. Here by an ordered cycle we just mean that it is a copy of some cycle C where V (C) has been assigned an ordering. Question 8.3. Let k ≥ 2 be a fixed integer and let s ∈ (0, 1). What is the minimum degree threshold that ensures an ordered n-vertex graph G contains a copy of some ordered cycle C of length at least sn and with χ < (C) = k?
The following question can be viewed as raising an ordered version of the well-known El-Zahar conjecture [6].
Question 8.4. Let k, t ≥ 2 be fixed integers and let C = {C 1 , . . . , C t } be a fixed family of t (not necessarily distinct) ordered cycles where χ < (C i ) = k for each i ∈ [t]. Set n i := |C i |. What is the minimum degree threshold that ensures an ordered graph G on n := n 1 + · · · + n t vertices contains vertex-disjoint copies of each cycle C 1 , . . . , C t that together cover the vertex set of G?
We suspect our approaches to regularity and absorbing will be useful for attacking this problem for large n. Variants of Question 8.4 (e.g., when C contains cycles of different interval chromatic number) would also be interesting to investigate.
Remark: Since this paper was submitted, the third author and Freschi [9] have asymptotically determined δ < (H, n) for all ordered graphs H with χ < (H) ≥ 3. Their approach both relies on tools from this paper, as well as introducing new ideas. They have also given an asymptotic solution to Question 8.2. Further, Hurley, Joos and Lang [15] have proven some very general tiling results, including a generalisation of the Kühn-Osthus tiling theorem that allows tiles to be different and to have size that grows with the size of the host graph. As pointed out in [15], it would be interesting to seek analogous results in the setting of ordered graphs too.