Singular Ramsey and Tur\'an numbers

We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $v\in V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $m\geq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Tur\'an number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop methods to estimate Rs$(F)$ and Ts$(n,F)$, present tight asymptotic bounds and exact results.


Introduction
In this paper we introduce a new type of Ramsey and Turán numbers, where the classical condition of the occurrence of a specified subgraph in an edgecolored complete graph is combined with restrictions on vertex degrees in the monochromatic host graph.

Brief survey on degree-constrained problems
The smallest particular case of Ramsey's theorem is that on six vertices every graph or its complement contains the triangle K 3 . Starting from here, Albertson [2] proved 1 that for n ≥ 6 in every 2-coloring of the edges of K n there is a monochromatic K 3 with two equal degrees. Inspired by this result several papers were written, see for example [3,5,6,8,11].
An obvious step after [2] is to try to generalize this result to other graphs and also to try to bound the difference between the maximum and minimum degree of the specified monochromatic subgraph. The efforts in the direction can be summarized as follows.
In [4] Albertson and Berman showed that K n can always be colored redblue in such a way that no red K 4 occurs and no blue K 2 has equal monochromatic degree at its two vertices. This shows that the phenomenon observed by Albertson is isolated and not extended to other graphs. But the authors of [4] also showed that for n ≥ 6 in every 2-coloring of K n there is a K 3 with spread of the degrees at most 5, where the spread of a sequence D =: {d 1 , . . . , d m } is defined as max{|d i − d j | | 1 ≤ i, j ≤ m}. An extension of their result is presented in [20].
In the papers [14,17] Chen, Erdős, Rousseau and Schelp developed the notion of spread explicitly and proved in [17] that every graph on at least k + 2 vertices contains at least k + 2 vertices whose degrees have spread at most k. This is a non-trivial extension of the popular observation that every graph with more than one vertex has two vertices of the same degree. From the quoted theorem the authors also proved among other things that for every graph G and every n ≥ R(G) (the classical Ramsey number) every 2-coloring of K n contains a monochromatic copy of G, whose vertex degrees in the host monochromatic graph have spread at most R(G) − 2, and that in a certain sense this upper bound is tight. An easy corollary is that the spread 5 from the Albertson-Berman result mentioned above can be reduced to 4 for n ≥ 6, which is best possible (as already noted in [17]).
Albertson [1] also introduced the corresponding Turán number, namely the maximum number of edges in a graph on n vertices having no copy of K m with all degrees equal, and presented an exact bound. (In an earlier paper [9] Caccetta, Erdős and Vijayan studied a Turán-type problem concerning the existence of a complete graph K m with large degrees.) A closely related subject is that of constant-degree independent sets, introduced by Albertson and Boutin [5], which was recently further developed by Caro, Hansberg and Pepper [11]. The latter considered various bounds on the constant-degree k-independent set in trees, forest, d-degenerate graphs and d-trees. Yet another direction concerns low-degree independent sets in planar graphs, developed by many authors and best presented in [6].
Further related notions are the so-called fair dominating sets (which actually are regular dominating sets, see Caro, Hansberg and Henning [10]), irregular independence number and irregular domination number (Borg,Caro and Fenech [8]), and the problem of monochromatic degree-monotone paths in 2-colorings of the edges of complete graphs (Caro, Yuster and Zarb [12]).

Singular Ramsey and Turán numbers
Albertson and Berman [4] presented edge 2-colorings of K n avoiding a monochromatic copy of G with all monochromatic degrees in K n equal. On the other hand, the opposite possibility of having a monochromatic copy of G with all its vertices having distinct monochromatic degrees in K n is very easy to exclude, by any decomposition of K n into two regular spanning graphs H and H. However, simultaneous exclusion of the two cases is impossible if n is large. This fact motivates our present study. Definition 1. Let k ≥ 1 be an integer. A sequence a 1 ≤ a 2 ≤ · · · ≤ a n of integers is called k-singular if either a 1 = · · · = a n or for every j = 1, . . . , n − 1, a j+1 − a j ≥ k. Also if a 1 , a 2 , . . . , a n are integers (repetitions are allowed), we say that they form a k-singular set if putting them in increasing order we obtain a k-singular sequence. (Hence, "set" may mean "multiset" in this particular context.) Definition 2. A subgraph H of graph G is called k-singular if the degree sequence of its vertices in G -where G is termed the host graph -forms a k-singular sequence.
For short, in case of k = 1, a 1-singular sequence is called singular sequence, and a 1-singular subgraph is called singular subgraph.
Let now F be a family of graphs.
Definition 3. The k-singular Ramsey number Rs(F , k) is defined as the smallest integer n such that in every 2-coloring of the edges of K m for any m ≥ n, one of the graphs induced by the color classes contains a k-singular member of F . Remark 4. If in a graph G the subsequence of degrees belonging to a set B of vertices is k-singular, then so does the subsequence belonging to B in the complement of G as well. Hence, in case of two colors, the vertex sets of k-singular subgraphs in color 1 coincide with those in color 2 (for any k ≥ 1).
In a similar flavor, as a little deviation, we also introduce a Turán-type function.
Definition 5. Given F , and a natural number k, the k-singular Turán number -as a function of the order n -denoted by Ts(n, F , k) is defined as the maximum number of edges in a graph G on n vertices that contains no k-singular copy of any F ∈ F . In particular, let Ts(n, q) be the maximum number of edges in a graph G of order n that contains no singular copy of K q .
For singular Ramsey numbers we shall use the simpler notation Rs(F ) = Rs(F , 1) for k = 1, and we write Rs(F ) for Rs({F }).
It is also natural to introduce non-diagonal and multicolored versions of Rs(F ). Definition 6. If F 1 and F 2 are two graphs, their singular Ramsey number Rs(F 1 , F 2 ) is the smallest n such that, for every m ≥ n, every 2-coloring of K m contains a singular copy of F 1 in the first color or a singular copy of F 2 in the second color. More generally, also for an integer s > 2, one may consider s families F 1 , . . . , F s of graphs and define the k-singular Ramsey number Rs(F 1 , . . . , F s , k) as the smallest integer n with the property that, for any m ≥ n, in every coloring of the edges of K m with s colors, there is an i (1 ≤ i ≤ s) such that the graph induced by the ith color class contains a k-singular 2 member of F i .

Remark 7.
(Non-monotonicity.) Let the number of colors be fixed. If every coloring of K n contains a monochromatic singular copy of some F ∈ F , still there is no guarantee that so does every coloring of K n+1 as well. This issue concerning (non-) monotonicity was observed already in the first papers by Albertson, and ever since; it is treated by imposing the condition for every m ≥ n in the definition of Rs(F ), rather than just taking the smallest n forcing a singular monochromatic F in every 2-coloring of K n . Remark 8. (Monotonicity Principle.) It is obvious -but will be applied at some point below -that the function Rs is monotone with respect to inclusion, for any fixed number of colors; for instance, if F 1 ⊆G 1 and F 2 ⊆G 2 , then Rs(F 1 , F 2 ) ≤ Rs(G 1 , G 2 ) holds.

Remark 9.
In the classical version of Ramsey and Turán numbers, isolated vertices are practically irrelevant, namely R(G ∪ mK 1 ) = max(R(G), m + |V (G)|); but this is not at all the case in the singular version. For instance, it can easily be shown (partly following also from some later observations) that for the graph G = P 3 ∪ K 1 -the path on 3 vertices plus an isolated vertexwe have Rs(P 3 ∪ K 1 ) = 10 while R(P 3 ∪ K 1 ) = 4, moreover Rs(P 3 ) = 5 and R(P 3 ) = 3. (Also, one may observe that Rs(3K 1 ) = 5 while R(3K 1 ) = 3.) Similarly, the Turán number of K 2 ∪ K 1 is zero for every n ≥ 3, but K 4 − e does not contain it as a singular subgraph, therefore Ts(4, K 2 ∪ K 1 , 1) = 5.
In this paper we will mostly consider Ramsey-type results for two colors, and develop a couple of methods suitable for determining the exact value of singular Ramsey numbers in both the diagonal and non-diagonal cases, provided that the specified graphs satisfy certain properties. We also present asymptotic estimates, and the k-singular version will be touched, too. In a section after the Ramsey-type results we provide tight asymptotics for the k-singular Turán number of a graph.

Our results
While the star graphs can be considered as the easiest infinite class of graphs concerning the classical Ramsey numbers (they almost admit a one-line proof), they turn out to be a bit complicated in the singular version. For this reason, although we present a complete solution, we do not discuss them earlier than in Section 5. Before that, we give some general lower and upper bounds (Section 2), describe some methods to derive tight estimates (Section 3), and determine exact results for all, but one, graphs with at most four vertices and edges, with the unique exception of C 4 (Section 4). Tight asymptotics for singular Turán numbers are given in Section 6. Some open problems are mentioned in the concluding section.

Terminology and notation
Particular graphs. We use standard notation P n and C n for the path and the cycle on n vertices; K p,q for the complete bipartite graph with p and q vertices in its classes; and mK 2 for the matching with m edges. The claw is the graph K 1,3 . The paw, which we abbreviate in formulas as P W , is the graph with four vertices and four edges obtained from K 3 by adding a pendant vertex (or from K 4 by deleting the edges of a P 3 ). The bull is the graph obtained from K 3 by adding two pendant vertices which are adjacent to two of its distinct vertices (a self-complementary graph with five vertices and five edges).
Vertex degrees. The degree of a vertex v in a graph G is denoted by d G (v), or simply d(v) if G is clear from the context. Minimum and maximum degree are denoted by δ(G) and ∆(G), respectively. A degree class consists of all vertices having the same degree; hence the degree classes partition V (G), and their number is equal to the number of distinct values which occur in the degree sequence of G. Given a vertex partition V 1 ∪ · · · ∪ V k = V (G), and a vertex v ∈ V i , the internal degree of v is the number of its neighbors inside V i , and its external degree is the number of its neighbors in V (G) \ V i .
Ramsey number. We denote the Ramsey number by R(F ), that is the smallest n such that in every 2-coloring of the edges of K n , one of the color classes contains a monochromatic member of F .
Substitution. Let H be a graph with k vertices v 1 , . . . , v k , and let F 1 , . . . , F k be k non-null graphs (F i = K 1 is allowed). The substitution of F 1 , . . . , F k into the "host graph" H, denoted by H[F 1 , . . . , F k ], is the graph whose vertex set is the disjoint union V (F 1 ) ∪ · · · ∪ V (F k ), each V (F i ) induces the graph F i itself, and two vertices x ∈ V (F i ) and y ∈ V (F j ) (i = j) are adjacent in H[F 1 , . . . , F k ] if and only if v i v j is an edge in H. In this construction we say that the graph F i is substituted for v i .
In a graph G = (V, E), the subgraph induced by a set Y ⊂V is denoted by G[Y ].

Singular Ramsey numbers: General bounds
We start with the following easy lemma.
Lemma 10. Every sequence of k(n − 1) 2 + 1 integers contains a k-singular subsequence of cardinality at least n.
Proof. Suppose we have no n equal elements in the sequence. Then we must have at least k(n − 1) + 1 elements of distinct values. Reorder them in increasing order, say a 1 < · · · < a k(n−1)+1 . Take the subsequence a jk+1 for j = 0, . . . , n − 1. Clearly this is a k-singular n-term sequence.
Theorem 11. For any two families F 1 , F 2 of graphs and every natural number k ≥ 1 the following general upper bound holds: Proof. Consider a 2-coloring of the edges of K m , for any m ≥ k(R(F 1 , F 2 )− 1) 2 + 1. Let G 1 and G 2 be the subgraphs obtained by the edges of color 1 and color 2, respectively. By Lemma 10 the sequence of degrees of the vertices of G 1 contains a k-singular subsequence of cardinality R(F 1 , F 2 ). The degrees of the corresponding vertices form a k-singular subsequence also in G 2 . Now consider the 2-colring induced on the complete graph on those R(F 1 , F 2 ) vertices. By definition there is either a monochromatic copy of a graph G ∈ F 1 in color 1 or of a graph H ∈ F 2 in color 2. Hence the degrees of G (in the first case) form a k-singular subsequence in the host graph G 1 or the degrees of H (in the second case) form a k-singular subsequence in the host graph G 2 . Thus a required k-singular subgraph occurs whenever An immediate corollary is: (i) For every graph G we have Rs(G) ≤ (R(G)−1) 2 +1, and also Rs(G, H) ≤ (R(G, H) − 1) 2 + 1 for any two graphs G and H.
(ii) Every 2-coloring of K k(n−1) 2 +1 contains a monochromatic k-singular tree of order at least n.
(ii) Consider the degree sequence in the graph induced by the edges colored 1. By Lemma 10 there is a k-singular subsequence of n degrees. Consider now the induced coloring on the complete graph K n whose vertices are those forming the k-singular sequence. Since every graph or its complement is connected, it follows that there is a connected monochromatic subgraph of order n whose degree sequence is k-singular in the host graph, and hence such a tree occurs.
Having proved a general upper bound, we next supply a general quadratic lower bound.
We will construct a graph H on (n − 1) 2 vertices whose vertex set V is partitioned into n − 1 subsets V 0 , . . . , V n−2 , each of cardinality n − 1, such that all vertices in V i have the same degree (i = 0, 1, . . . , n − 2) but vertices from distinct subsets have distinct degrees. Then clearly no copy of G in H and H can be singular, as it must take at least two vertices in the same class and at least two vertices in distinct classes.
If n−1 is even, then we simply insert any i-regular graph inside V i . (Such graphs exist, e.g. by taking i perfect matchings from any 1-factorization of K n−1 .) If n − 1 is odd, then depending on residue modulo 4, one of the sequences 0, 1, . . . , n − 2 and 1, 2, . . . , n − 1 contains an even number of odd terms. If it is 0, 1, . . . , n−2, then we insert a regular graph of degree 2·⌊ i 2 ⌋ inside V i (e.g., the union of ⌊i/2⌋ edge-disjoint Hamiltonian cycles of K n−1 ). Moreover we insert a perfect matching between V 1 and V 3 , between V 5 and V 7 , ..., between V n−5 and V n−3 . Else, if it is 1, 2, . . . , n − 1, then we insert a regular graph of degree 2 · ⌊ i+1 2 ⌋ inside V i (e.g., the union of ⌊ i+1 2 ⌋ edge-disjoint Hamiltonian cycles of K n−1 ) and take a perfect matching between V 0 and V 2 , between V 4 and V 6 , ..., between V n−4 and V n−2 .
These graphs satisfy the requirements, proving the lower bound for all n.

Remark 14.
An alternative proof -which also works in the k-singular case for k ≥ 2 -can be obtained from the Erdős-Gallai characterization of graphical sequences. We note that for some combinations of k and n (both even) an analogous construction with k(n−1) 2 vertices is not possible, because a graph cannot have an odd number of odd-degree vertices.
In particular, the following bounds are obtained from the above estimates.
Corollary 15. If G is a graph of order n ≥ 3, then If G is a class of graphs in which R(G) is a linear function of |V (G)| over all graphs G ∈ G, then the growth order of both estimates in Corollary 15 is quadratic in n. In particular, applying the theorem of [15] on the Ramsey numbers of graphs with bounded maximum degree, we obtain: Theorem 16. Let G be the class of graphs with bounded degree ∆ fixed. Then for all G ∈ G of order n we have Rs(G) = Θ(n 2 ), as n → ∞.
We conclude this section with a sufficient condition ensuring that the lower bound in Corollary 15 holds with equality. This result also exhibits a significant difference between the classical and the singular versions of Ramsey numbers concerning the role of isolated vertices. Proof. We only have to prove that (|V (G)| − 1) 2 + 1 is an upper bound on Rs(G). If n ≥ (|V (G)|−1) 2 +1, then in every 2-coloring of K n the subgraph of color 1 contains a singular subgraph, say G * , on |V (G)| = |V (H)|+m ≥ R(H) vertices. Thus, a singular monochromatic copy of H occurs, either in color 1 or in color 2, which can be supplemented to a singular copy of G because the m isolated vertices put no restriction on the color distribution in the rest of G * .

Some methods
Assume that a graph G has been fixed, for which we wish to find estimates on Rs(G). We say that a graph F is G-free if F does not contain any subgraph isomorphic to G. Moreover, let us call F an R-graph for G if both F and F are G-free. Analogously, we say that F is an SR-graph ('S' standing for 'singular') for G if neither F nor F contains a singular subgraph isomorphic to G.
Lower bounds on Rs(G) will be obtained by constructing SR-graphs from several (smaller) R-graphs. We call this the technique of canonical colorings.
Possible different approaches will be described in the next two subsections, and a kind of combination of them afterwards.
The fourth subsection presents a method to derive upper bounds when some favorable information concerning the structure of R-graphs of order R(G) − 1 is available. This approach will lead to exact results in several cases. Finally we mention another approach to upper bounds, based on vertex degrees.

Non-regular Canonical Coloring, NRCC
This approach is useful when 'large' R-graphs are not regular. For instance, the claw K 1,3 and its complement K 3 ∪ K 1 are the two R-graphs of order 4 for G = 2K 2 , and also for P 4 , but neither of them is regular. We apply this method in Section 4.2.
Let G be a graph on n vertices, and let t ≤ n − 1. Consider t copies of (not necessarily isomorphic) R-graphs over mutually disjoint vertex sets V 1 , . . . , V t . Suppose that we can insert edges between the vertex classes (but not inside them) to obtain a graph H with the following properties: 1. In each vertex class V i (i = 1, . . . , t) all the degrees d H (v) are equal.
2. Degrees of vertices belonging to distinct vertex classes are distinct.
Lemma 18. With the assumptions above, we have Proof. In such a case H and H have exactly t classes of distinct degrees and t ≤ n − 1, hence no copy of G with all degrees distinct is possible (there are too few distinct degree classes). Also, since each set V i induces an R-graph in H, no copy of G with all degrees equal is possible as it should be contained in a unique degree class. Hence H is an SR-graph, showing Rs(G) ≥ |V (H)|+1.

Regular Canonical Coloring, RCC
We can apply this approach when there exist 'large' R-graphs which are regular. (The first classical example is G = K 3 whose unique largest Rgraph is C 5 .) We apply this method in Section 4.3.
Let F be an R-graph on q vertices v 1 . . . , v q , and let H 1 , . . . , H q be q further R-graphs. Denote by H = F [H 1 , . . . , H q ] the graph obtained by taking the vertex-disjoint copies of H 1 , . . . , H q and making all the vertices of H i adjacent to all the vertices of H j if and only if the vertices v i and v j are adjacent in F .
Suppose that H has the following properties: Lemma 19. With the assumptions above, we have Proof. Observe first that since all vertices of H i are connected to the same vertices outside H i and also have the same degree inside H i , it follows that H i is a regular subgraph in H (hence the name Regular Canonical coloring).
Since H i and its complement H i are G-free, property 2 implies that there is no copy of G with all degrees equal. If there was a copy of G with all degrees distinct in H, then no H i would contain more than one vertex from G. Hence, by the construction, there would be a copy of G in F , but this is impossible because F and F are G-free.
Thus H is an SR-graph for G, and hence Rs(G) ≥ |V (H)| + 1.

A mixed construction
We apply this method in Sections 4.3 and 4.4. The construction starts with a graph H such that H contains no singular G 1 and H contains no singular G 2 . Partition V (H) into some number of subsets, say V (H) = X 1 ∪ · · · ∪ X k ; many of those X i may also be singletons. As a generalization of substitution, we replace those X i with mutually vertexdisjoint graphs Q 1 , . . . , Q k such that each Q i is regular, G 1 -free, and Q i is G 2 -free. The plan is to create a graph F whose degree classes are the sets V (Q i ), using the structure of H. If X i consists of q i vertices from H, then we partition V (Q i ) into q i subsets. The vertices in the j th part of Q i are completely adjacent to those classes Q ℓ which correspond to the neighbors of the j th vertex of X i in H. (In particular, if X i and X ℓ are singletons adjacent vertices, then we take complete bipartite adjacency between Q i and Q ℓ .) A delicate detail in this approach is to ensure that two vertices have the same degree if and only if they are in the same Q i . This needs a careful choice of the orders |V (Q i )|, the internal degree of each Q i , and also the sizes of the partition classes inside Q i .

Ramsey-stable graphs
The tool described in this subsection will turn out to be substantial, in the proofs of upper bounds in several results below.
Let G be a given graph for which we wish to determine or estimate the value of Rs(G). Consider an R-graph H for G, say with k vertices v 1 , . . . , v k . Let N i denote the set of vertices adjacent to v i (the neighborhood of v i ).
Definition 20. We call H a Ramsey-stable graph for G if, for each 1 ≤ i ≤ k, the unique way to obtain an R-graph of order k, in which H −v i is an induced subgraph, is to join a new vertex to all vertices of N i , and not to join it to any other vertex of H − v i . Ramsey-stable graphs for a pair (G 1 , G 2 ) of graphs can be defined analogously.
Example 21. The 5-cycle is Ramsey-stable for K 3 , and also for K 1,3 , because the only way to extend P 4 to an R-graph for K 3 , or for K 1,3 , is to join a new vertex to the two ends of P 4 .
Remark 22. More generally than the previous example, if we know that all n-vertex R-graphs for a given G are regular, then every R-graph H of order n is Ramsey-stable for G because exactly the vertices of minimum degree in H − v have to be joined by an edge to the new vertex.
Assume that F is an SR-graph for a given graph G, and that the degree sequence of F contains precisely k distinct values. We partition V (F ) into the degree classes V 1 , . . . , V k . Pick one (any) vertex v i from each class V i , and denote by H the graph induced by The significance of Ramsey-stable graphs is shown by the following lemma, which will be crucial in several proofs later on. As a side product, it also implies that if a suitable choice of {v 1 , . . . , v k } gives us a Ramsey-stable H, then all possible choices of the v i ∈ V i (i = 1, . . . , k) yield the same H.
Lemma 23. (Regular Substitution Lemma.) Let F, G, H be graphs as above.
If H is Ramsey-stable for G, then F is obtained from H by substituting a regular R-graph for each vertex v i of H. The same structure is valid when H is Ramsey-stable for a pair (G 1 , G 2 ).
Proof. Assume that H is Ramsey-stable for G; the case of (G 1 , G 2 ) can be handled in exactly the same way. Then for any i, replacing the vertex V (H)∩ V i with any v ∈ V i , the neighborhood remains the same, by assumption. Hence every v j ∈ N i (which has been taken from the degree class V j ) is completely adjacent to V i . This is true also when we view the edge v i v j from the other side, from v j ; therefore v i -and each of its replacement vertices, v ∈ V i -is adjacent to the entire V j . Consequently, for each edge v i v j of H, the edges between V i and V j in F form a complete bipartite graph spanning On the other hand, by the analogous argument for the non-edges of H, we see that if v i v j is not an edge in H, then there are no edges between V i and V j in F . Thus, F is generated by the operation of substitution. As a quantitative consequence, the external degrees of vertices in any one V i are all equal.
Equal external degrees imply for a degree class that the internal degrees must also be equal. This implies regularity inside each V i .

Vertex degrees
In some cases the following approach is useful in deriving upper bounds on Rs(G). We apply it in Section 4.3.
Lemma 24. If, for a given graph G, every SR-graph of order n has minimum degree δ, then there can be at most n − 2δ degree classes.
Proof. Consider any SR-graph F of order n, and let k denote the number of its degree classes. Then, concerning the minimum and maximum degree we have Typically one can use this in the way that if n is large then an SRgraph should have not only large minimum degree but also a large number of degree classes, from which a contradiction is derived to the above inequality, concluding that Rs(G) ≤ n.

Exact results on Rs(G) for small graphs
The smallest nontrivial cases are the path P 3 = K 1,2 and its subgraphs; they allow a simple solution for k-singular Ramsey numbers for all k, which we present in the first subsection. In this way K 3 remains the unique graph G of order three for which we do not know Rs(G, k) over the entire range of k.
All other subsections of this section deal with the case k = 1 for small graphs, determining Rs for every graph with at most four vertices and at most four edges, except for C 4 where we have a non-trivial lower bound. This also includes small star graphs (the claw K 1,3 , and the K 1,2 which is treated under the name P 3 ); a general theorem for stars will be presented in Section 5.

The path P 3 for general k of singularity
Proof. Clearly, by Theorem 11 above we get Rs(P 3 , k) ≤ 4k+1 as R(P 3 ) = 3.
For the lower bound consider the graph H(k) on 4k vertices defined as follows: In this graph, which treats the lower bound for the three graphs 3K 1 , K 2 ∪ K 1 , P 3 together, every degree between 1 and 2k is repeated exactly twice, i.e. no triple can have equal degrees. Also there cannot occur any ksingular subgraph of order three, because this would require that ∆(H(k)) − δ(H(k)) ≥ 2k, however in H(k) and hence also in its complement the difference is just 2k − 1.
For the lower bound on Rs(2K 2 ) we construct an SR-graph on 12 vertices.
We complete these vertex classes to a graph G (color 1) such that all degrees in V 1 are 7, all degrees in V 2 are 5, and all degrees in V 3 are 4. Once this shall be done, there will be no copy of 2K 2 with all degrees equal in G and neither in G because each V i induces K 3 ∪ K 1 in G and K 1,3 in G. Also there will be no 2K 2 with all degrees distinct since this would require four different degrees, while in both G and G there are only three. We shall do the construction step by step.
For the upper bound on Rs(P 4 ) we will apply the Regular Substitution Lemma. On four vertices precisely two graphs are R-graphs for P 4 : the claw K 1,3 and its complement, the triangle K 3 with an isolated vertex. On five vertices every edge 2-coloring contains a monochromatic P 4 . Observe that each of K 1,3 and K 3 ∪ K 1 is a Ramsey-stable graph for Rs(P 4 ), because a 3-vertex subgraph with zero or two edges is extendable only to the claw, whereas that with one or three edges is extendable only to the triangle; either extension is unique also concerning the set of neighbors of the new vertex.
Suppose now for a contradiction that there exists an SR-graph F for P 4 on at least 13 vertices. There can be at most four degree classes in F , each on at most four vertices. It follows that there are precisely four vertex classes. Due to Lemma 23, each degree class should induce a regular R-graph; but this is impossible for a class with four vertices, which must occur if |V (F )| > 12. This contradiction completes the proof.
Since the host graph and also the subgraphs substituted for the degree classes are K 3 -free and K 1,3 -free, no singular K 3 or K 1,3 occurs.
Proof of Upper Bound 22. Since R(K 3 ) = R(K 1,3 ) = 6, we infer from Theorem 11 that Rs(K 3 ) ≤ 26 as well as Rs(K 1,3 ) ≤ 26. So we have to cover the cases n = 22, 23, 24, 25, to show that a singular triangle and a singular claw necessarily occurs in each case.
For 22 = 5 + 5 + 4 + 4 + 4 note further that a 4-element V i must induce the 2-regular C 4 or the 1-regular 2K 2 . Thus, all internal degrees are between 1 and 2, and all external degrees are between 8 and 10, leaving room for no more than four degree classes while we should have five of them. For this reason, the case 22 = 5 + 5 + 4 + 4 + 4 cannot occur, and the same argument excludes 23 = 5 + 5 + 5 + 4 + 4.
The only case that remains is 22 = 5 + 5 + 5 + 4 + 3. External degree 7 can only occur for a 5-element class which has internal degree 2. External degree 8 can only occur for a 4-element or a 5-element class, both having internal degree at least 1. All other possibilities yield external degrees at least 9, thus δ(F ) ≥ 9, and we can conclude as in the first proof that ∆(F ) ≥ 13 should hold, from which we arrive at the contradiction δ(F ) ≤ 8.
It turns out that the non-diagonal singular Ramsey number Rs(K 3 , K 1,3 ) is bigger.

Proof of Lower Bound 29.
We construct a graph F on 28 vertices, without singular triangles, whose complement F does not contain any singular claws. This F will have k = 5 degree classes V 1 , . . . , V 5 , where |V 1 | = |V 3 | = |V 5 | = 6 and each of those classes induces K 3,3 , while |V 2 | = |V 4 | = 5 and both classes induce C 5 . Hence each degree class is internally regular, with no K 3 in it, and no K 1,3 in the complementary graph.
One can observe that every triangle of F contains two vertices in the same V i and one vertex in another class, hence no singular triangles occur. Similarly the complement of F contains no singular claws. Thus F satisfies all requirements and yields Rs(K 3 , K 1,3 ) ≥ 29.
Concerning the upper bound we first observe some structural properties of the graphs which are R-graphs for (K 3 , K 1,3 ). Claim 1. We have R(K 3 , K 1,3 ) = 7, and the unique R-graph of order 6 is K 3,3 .
Proof. Observe that K 3,3 is the unique triangle-free graph of order 6 whose minimum degree is at least 3. On the other hand, if the minimum degree is smaller than 3, then the complement contains K 1,3 .
Claim 2. On five vertices there are precisely two graphs H -namely C 5 and K 2,3 -such that H is triangle-free and H is K 1,3 -free. The first one, C 5 , is a Ramsey-stable graph for (K 3 , K 1,3 ).
Proof. All vertex degrees must be at least 2 (otherwise H contains K 1,3 ) and at most 3 (otherwise H contains K 3 or H contains K 1,3 ). If H is 2regular, then H ∼ = C 5 . In the remaining case assume that d(v) = 3. The three neighbors of v must be mutually non-adjacent, otherwise K 3 ⊂H; and all of them have to be adjacent to the fifth vertex, since δ(H) ≥ 2. No further edges can occur, hence H ∼ = K 2,3 in this case. Since no other R-graphs are possible, and P 4 is not an induced subgraph of K 2,3 , it is clear that C 5 is Ramsey-stable.
Claim 3. Among the regular four-vertex graphs H there are precisely two -namely C 4 and 2K 2 -such that H is triangle-free and H is K 1,3 -free.
Proof. The other two regular graphs of order 4 are K 4 which contains K 3 , and 4K 1 whose complement contains K 1,3 .

Proof of Upper Bound 29.
Let F be an SR-graph for (K 3 , K 1,3 ), say on n := Rs(K 3 , K 1,3 )−1 vertices; we need to prove that n ≤ 28. We see from Claim 1 that F has at most six degree classes V 1 , . . . , V k , and |V i | ≤ 6 holds for each of them.

Case 1: Four vertex classes.
This case is obvious: since |V i | ≤ 6 holds for all i, we cannot have more than 24 vertices.

Case 2: Six vertex classes.
Picking one vertex v i from each vertex class V i we obtain an R-graph H of order 6. Due to Claim 1, we have H ∼ = K 3,3 , which is Ramsey-stable. The Regular Substitution Lemma implies that F is obtained by substituting regular R-graphs for the vertices of H; the possible subgraphs with more than three vertices are listed in Claims 1, 2, and 3 (and K 2,3 is excluded). Let us denote the subgraphs substituted into the partite sets by Q 1 , Q 2 , Q 3 and R 1 , R 2 , R 3 ; and let their respective orders be q 1 , q 2 , q 3 , r 1 , r 2 , r 3 . Also let us write d(q 1 ), d(q 2 ), d(q 3 ), d(r 1 ), d(r 2 ), d(r 3 ) for their internal degrees. We fix an indexing such that d(q 1 ) ≥ d(q 2 ) ≥ d(q 3 ) and d(r 1 ) ≥ d(r 2 ) ≥ d(r 3 ). Note that all these d are between 0 and 3 (and 0 can occur only if the substituted graph has at most three vertices).
Denoting q = q 1 + q 2 + q 3 and r = r 1 + r 2 + r 3 , the degree set of F is with six mututally distinct values. In particular, we must have strict inequalities d(q 1 ) > d(q 2 ) > d(q 3 ) and d(r 1 ) > d(r 2 ) > d(r 3 ). It follows on each side of K 3,3 that each of K 3,3 and C 5 can be substituted only once, which implies max(q, r) ≤ 15. Moreover, assuming q ≥ r the degrees cannot be smaller than r and cannot be larger than q + 3, hence the presence of six distinct degrees implies q + 3 ≥ r + 5, i.e. r ≤ q − 2 ≤ 13. Thus n = p + r ≤ 28.
As above, we pick one (any) vertex v i from each V i (1 ≤ i ≤ 5), and consider the graph H induced by them in F . Due to Claim 2, this H must be C 5 or K 2,3 . Since C 5 is Ramsey-stable, the proof for it is easy. Indeed, as above, the Regular Substitution Lemma implies that F is obtained by substituting regular R-graphs for the vertices. But n = 29 or n = 30 would imply that along the 5-cycle four consecutive substitutions would be K 3,3 . The two middle ones of them would have external degree 12, internal degree 3, total degree 15, contradicting the assumption that they form distinct degree classes.
Hence, from now on we assume that H ∼ = K 2,3 . Re-label the indices, if necessary, so that the 2-element class of K 2,3 is {v 1 , v 2 } and the 3-element class is {v 3 , v 4 , v 5 }. Although K 2,3 is not Ramsey-stable, vertices v 1 and v 2 have the property that replacing any one of them with a vertex from its class, we must obtain again a K 2,3 , which implies that {v 3 , v 4 , v 5 } is completely adjacent to V 1 ∪ V 2 . Due to the exclusion of singular K 3 , this also forces that V 1 and V 2 are completely non-adjacent.
For a vertex v ∈ V i from V 3 ∪ V 4 ∪ V 5 there can occur two situations: v is adjacent either to v 1 and v 2 -in which case we say that v is in a stable position -or to the two vertices of {v 3 , v 4 , v 5 }\{v i }.
If all v ∈ V 3 ∪ V 4 ∪ V 5 are in a stable position, then we have complete adjacency between V 1 ∪ V 2 and V 3 ∪ V 4 ∪ V 5 , moreover no edges can occur between V 3 and V 4 , between V 3 and V 5 , and between V 4 and V 5 , and also between V 1 and V 2 either. This yields regular external degrees for each V i . Hence the internal degrees of V 3 , V 4 , and V 5 must be regular and mutually distinct, as well as those in V 1 and V 2 , what implies that |V 1 ∪V 2 | ≤ 6+5 = 11 and |V 3 ∪ V 4 ∪ V 5 | ≤ 6 + 5 + 4 = 15, thus n ≤ 26.
The occurrance of vertices in non-stable position requires a little more structural analysis. For this, suppose that a v ∈ V 5 is adjacent to v 3 and v 4 , instead of v 1 and v 2 . Since F contains no singular claws, and v ∈ V 5 already has non-neighbors in V 1 and V 2 , all vertices of V 3 ∪ V 4 are adjacent to v. Then no edges can occur between V 3 and V 4 , for otherwise F would contain a singular K 3 . Similarly, v has no neighbors in V 1 ∪ V 2 , because such a neighbor and v would form a singular K 3 with v 3 (and also with v 4 ).
The non-adjacency of V 3 and V 4 also implies that all vertices in V 3 ∪ V 4 are in a stable position. Thus, we have the following structure: • there is complete adjacency between V 1 ∪ V 2 and V 3 ∪ V 4 ; • no other edges occur between any V i and V j for 1 ≤ i < j ≤ 5.

The paw graph
As another small graph, we determine the singular Ramsey number of the paw, that is a triangle with a pendant edge. Its Ramsey number is R(P W ) = 7. Let us first summarize some facts about the R-graphs.
Lemma 29. For the paw graph, (i) every graph on at most three vertices is an R-graph, and among them, the regular ones are K 3 and ist complement; (ii) on four vertices there are two regular R-graphs, the 1-regular 2K 2 and the 2-regular C 4 ; (iii) on five vertices there are three R-graphs, namely K 2 ∪ K 3 and its complement K 2,3 which are non-regular, and the 2-regular C 5 ; (iv) on six vertices there are two R-graphs, the 2-regular 2K 3 and its 3regular complement, K 3,3 .
Proof. Parts (i) and (ii) are obvious. For (iii) one may note that C 5 is the unique R-graph for K 3 on five vertices, and of course it is an R-graph for the paw, too. If G contains a triangle and is an R-graph for the paw, then the triangle is a connected component. This implies that on five vertices the complement of G must contain K 2,3 , and on six vertices the complement must contain K 3,3 . Then there cannot be any further edges in G, hence The quadratic formula yields the upper bound 37 on Rs(P W ), but in fact the exact value is much smaller.

Proof of Lower Bound 31.
We construct a graph F of order 30 which is an SR-graph for the paw. It will have five degree classes V 1 , . . . , V 5 , each of cardinality 6. The degree classes induce R-graphs: (One may verify in the proof below that it would be equally fine to take F [V 5 ] ∼ = K 3,3 .) Further, we partition V 5 as V 5 = V ′ ∪ V ′′ , with |V ′ | = 2 and |V ′′ | = 4.
We make complete adjacencies between any two of the three sets V 1 , V 2 , V ′ ; and also between any two of V 3 , V 4 , V ′′ . There are no further adjacencies; i.e., the only edges between This F contains no regular paw, because the degree classes are paw-free; and it has no irregular paw either, because omitting the internal edges of the degree classes (which edges certainly cannot occur in any irregular subgraph) we obtain a graph which is generated by substituting independent sets into 2K 3 . Now we have the following degrees: • in V 1 : external 6 + 2 = 8, internal 2, total 10; • in V 2 : external 6 + 2 = 8, internal 3, total 11; • in V 3 : external 6 + 4 = 10, internal 2, total 12; • in V 4 : external 6 + 4 = 10, internal 3, total 13; • in V 5 : external 6 + 6 = 12, internal 2, total 14.
Hence, F satisfies all requirements and yields Rs(P W ) > 30.
Proof of Upper Bound 31. Suppose for a contradiction that there exists an SR-graph F on at least 31 vetices. We know that each degree class contains at most six vertices, therefore we have exactly six degree classes V 1 , . . . , V 6 . Picking one vertex v i from each V i , we get an R-graph, say H, of order 6, which must be either 2K 3 or K 3,3 , due to Lemma 29(iv). Turning to the complement of F if necessary, we may assume without loss of generality that H ∼ = 2K 3 .
Since 2K 3 is Ramsey-stable, we see from the Regular Substitution Lemma that F is obtained by substiting regular R-graphs for the vertices of H. We are going to analyze the feasible substitutions which create three distinct degree classes for each of the two components. We shall see that it is not possible to have more than 16 vertices in a component, there is just one way to obtain 16, and there are exactly two ways to obtain 15. Hence only the combinations 32 = 16 + 16 and 31 = 16 + 15 would yield n > 30, but the argument below will show that each of them would force equal degrees to at least two of the V i , which contradicts the definition of degree class.
18: The unique feasible partition is 18 = 6 + 6 + 6. But then two of the degree classes induce the same R-graph (2K 3 or K 3,3 ), therefore they have the same degree in F , a contradiction. 17: The unique feasible partition is 17 = 6 + 6 + 5. Then the vertices in both 6-classes have external degree 11. In order that they have different degrees in F , one of them must induce K 3,3 and the other induce 2K 3 . Then their degrees in F are 14 and 13, respectively. However, the class of five vertices has external degree 12 and internal degree 2, yielding total 14, which is not feasible. 16: The two feasible partitions are 16 = 6 + 6 + 4 and 16 = 6 + 5 + 5. The latter is easy to exclude, because both 5-classes have internal degree 2 and external degree 11. Concerning 16 = 6 + 6 + 4 we see that the two subgraphs for '6' must have distinct internal degrees, hence one of them is 2K 3 , the other is K 3,3 . Both have external degree 6 + 4 = 10, hence the vertex degrees are 12 and 13, respectively. This implies that the subgraph for '4', which has external degree 12, must be C 4 because 2K 2 with internal degree 1 would repeat the degree 13. Thus the degrees necessarily are 12, 13, 14. Of course, this cannot occur on more than one triangle; i.e., the case n = 32 = 16 + 16 is impossible. We can immediately exclude the last one because '5' necessarily means C 5 with internal degree 2, hence in a substitution of the type 5 + 5 + 5 the graph would be regular of degree 12. Concerning 15 = 6 + 6 + 3 -similarly to the case of 16 = 6 + 6 + 4 -we see that 2K 3 and K 3,3 have to be substituted for 6 + 6, yielding vertex degrees 11 and 12. For '3' we have external degree 12, hence 3K 1 is not an alternative, we have to substitute the other regular graph, K 3 , which has internal degree 2. In this way we obtain the degrees 11,12,14 which cannot be coupled with the case (12,13,14) of 16 = 6 + 6 + 4.
In 15 = 6+5+4 the '5' class means C 5 with internal degree 2 and external degree 10, i.e. degree 12 in F . Therefore the '4' class with external degree 11 must be C 4 with internal degree 2 and total degree 13. The external degree for '6' is 9, hence internal degree 3 is infeasible, thus we have to substitute 2K 3 which leads to degree 11 and in this way we obtain the degree set 11, 12, 13. From this, it is clear that 16 + 15 cannot occur, and even 15 + 15 would be impossible. (In fact, degree 12 appears in all the three types above, and any two types have two values in common.) Remark 31. The construction on 30 vertices is another example of the mixed principle as described in Section 3.3. Here we start from the graph H = 2K 3 + e, two vertex-disjoint triangles connected by just one edge e. Although this H is not paw-free, still does not contain a singular paw; and its complement H ∼ = K 3,3 − e is paw-free. Then the two ends of the edge e can be viewed together as one partition class, while the other classes are singletons. Each end of e has two neighbors in H and this yields two neighbor classes for the corresponding subsets after substitution. In case of the paw, two classes of order 6 with identical neighborhood may occur because their internal degree can (and should) be distinct.

The 4-cycle C 4
In case of C 4 , which seems most problematic among the small graphs, we can derive lower and upper bounds which are quite close to each other, but still the exact value of Rs(C 4 ) is unknown.
Note that the 4-cycle has R(C 4 ) = 6, and its two R-graphs of order 5 are C 5 and the bull. Neither of them is Ramsey-stable. Indeed, removing a vertex from C 5 we obtain P 4 , which is extendable not only to C 5 itself, but also to the bull. Similarly, removing the degree-2 vertex from the bull we obtain P 4 which is extendable to C 5 . Moreover, the removal of a pendant vertex from the bull yields the paw, which can be extended to the bull in two different ways. Also, removing a vertex of degree 3 we obtain P 3 ∪ K 1 , whose extension to the bull fixes an edge to the isolated vertex, and another edge to the middle of P 3 , but the last edge can go to either end of P 3 .
Proof. The upper bound is a consequence of Corollary 12. For the lower bound we construct an SR-graph on 23 vertices. Let us take the bull as host graph H, labeling its vertices as v 1 , . . . , v 5 where {v 1 , v 2 , v 3 } induces a triangle, and the two pendant edges are v 1 v 4 and v 3 v 5 . Let us substitute graphs F i for v i such that F 1 ∼ = K 3 and F i ∼ = C 5 for all 2 ≤ i ≤ 5. All internal degrees are equal to 2, and the external degrees are 15 in F 1 , 8 in F 2 , 13 in F 3 , 3 in F 4 , and 5 in F 5 . Neither F nor its complement contains any singular copy of C 4 , hence Rs(C 4 ) ≥ 24.

Small graphs with isolates
In this last of the subsections devoted to small graphs we give the values for those graphs of order four which have isolated vertices. There are three such graphs: K 2 ∪ 2K 1 , P 3 ∪ K 1 , and K 3 ∪ K 1 . Note that some lower bounds can easily be obtained from above: • Theorem 13 (with reference also to Remark 8) implies • The construction of Theorem 27 yields We prove that these bounds are tight.
Proof. In every graph G with 10 vertices there exists a singular subgraph of order four. It necessarily contains a P 3 or its complement, which can be extended to a singular P 3 ∪ K 1 . Thus Rs(P 3 ∪ K 1 ) ≤ 10.
Proof. Suppose for a contradiction that F is an SR-graph of order n ≥ 22 for K 3 ∪ K 1 . We know that a singular K 3 occurs in F (or in its complement), say it has the vertices v 1 , v 2 , v 3 . If the degrees of this K 3 are all distinct, say d 1 < d 2 < d 3 , then it would be extendable to a singular K 3 ∪ K 1 unless all vertices of F have their degree from {d 1 , d 2 , d 3 }. But then a degree class would have at least eight vertices, so that F would contain even a singular Hence suppose that the three vertices of any singular K 3 have the same degree in F . Since R(K 3 ) = 6, there can be at most five degree classes, and we easily find a singular K 3 ∪K 1 unless all degree classes have at most five vertices and the degree class(es) inducing a triangle have exactly three vertices. In particular, {v 1 , v 2 , v 3 } itself is a degree class, moreover its complementary 19 or more vertices form only four degree classes. Now we can only have the following possibilities: • n = 22 = 3 + 4 + 5 + 5 + 5, • n = 23 = 3 + 5 + 5 + 5 + 5.
And then the degree counting method in the proof of Theorem 27 can be repeated for these two cases without any changes, leading to the contradiction δ(F ) ≥ 9 for n = 22 and δ(F ) ≥ 10 for n = 23.

Stars of any size
The star with s edges, K 1,s , is an easy case concerning Ramsey numbers; cf. e.g. Section 5.5 of [19]: • if s is odd, then R(K 1,s ) = 2s, and the extremal R-graphs are precisely the (s − 1)-regular graphs of order 2s − 1; • if s is even, then R(K 1,s ) = 2s − 1, and the extremal R-graphs are the graphs of order 2s−2 with minimum degree at least s−2 and maximum degree at most s − 1. In particular, the largest regular R-graphs are those graphs of order 2s − 2 which are (s − 2)-regular or (s − 1)-regular.
It turns out that the parity of s is essential also with respect to Rs. The case of even s is simpler, the quadratic upper bound always is tight.
Proof. The upper bound (2s − 2) 2 + 1 follows from Corollary 12. For the same lower bound we construct an RS-graph F of order (2s − 2) 2 on vertex set where all sets A i and B i are mutually disjoint, each of cardinality 2s − 2.
The edges of F are defined as follows: • each A i induces an (s − 2)-regular graph; • each B i induces an (s − 1)-regular graph; • there is no edge between A i and A j for i = j; • there is no edge between B i and B j for i = j; • every A i and B j are completely adjacent for i = j; • the sets A i and B i are adjacent by a (2i)-regular bipartite graph, for all 1 ≤ i ≤ s − 1.
Then, both in F and in F , each vertex v is adjacent to at most s − 1 vertices of distinct degrees different also from the degree of v; and to at most s − 1 vertices whose degree is equal to that of v. Thus, F is an RS-graph for K 1,s .
The case of odd s is more complicated. The quadratic upper bound is never attained, although the singular Ramsey number is not far from it. For the tightness of the lower bound we give two very different constructions, with the purpose to indicate that -contrary to R(K 1,s ) -the extremal graphs for Rs(K 1,s ) may be quite hard to characterize.
Theorem 36. If s is odd, then Proof of the Upper Bound. Suppose for a contradiction that there exists an SR-graph F of order at least (2s − 1)(2s − 2) + 2 for K 1,s . Denote its degree classes by V 1 , . . . , V m . We know that m ≤ 2s−1, and also |V i | ≤ 2s−1 for all 1 ≤ i ≤ m. Hence m = 2s − 1 must hold. Since all R-graphs of order 2s − 1 are regular, all of them are Ramsey-stable, due to Remark 22. Thus, by the Regular Substitution Lemma, each V i induces a regular R-graph, and two distinct V i , V j are either completely adjacent or completely nonadjacent, From this we obtain that the structure of adjacencies between the degree classes is an (s − 1)-regular graph of order 2s − 1, therefore • every external degree is at most (s−1)(2s−1), and every internal degree is at most s−1, therefore the maximum degree of F is at most 2s(s−1) and the minimum degree cannot be larger than 2s(s − 1) − (2s − 2) = 2(s − 1) 2 .
We use the symbol G d p to denote any d-regular graph on p vertices. Such graphs exist whenever p > d ≥ 0 and pd is even. In the construction below, the actual structure of a G d p will be irrelevant, one may take different graphs for different appearances of the same pair (p, d). Using the notation V i and G i in the sense as above, we now define: • for every i in the range 2q ≤ i ≤ 6q we take |V i | = 8q + 1, and let each G i be a G s−1 2s−1 ; • with the only one exception of V 8q , for all 1 ≤ i ≤ q we take |V 4q±(2q+2i) | = 8q, and let each G i be a G s−2 2s−2 ; • with the only one exception of V 2q−1 , for all 1 ≤ i ≤ q we take |V 4q±(2q+2i−1) | = 8q − 2, and let each G i be a G s−2 2s−4 ; • for the two exceptional cases we take |V 8q | = 8q − 2 with G 8q = G s−3 The maximum degree occurs at the vertices of V 4q : they have internal degree s − 1, external degree (s − 1)(2s − 1), and total degree 2s(s − 1). Relevant parameters of vertices in the other degree classes are summarized in Table 2.
One can check that each V i has a distinct degree, and consequently we obtained an SR-graph of maximum order.
Notations SZ, ID, ED, TD are the same as in Table 1.

Asymptotics for singular Turán numbers
In this section we present estimates on the singular Turán numbers Ts(n, F ), and compare them to the classical Turán number ex(n, K s ), which is the maximum number of edges in a graph of order n not containing a complete subgraph of order s. Assume 3 that F has order p := |V (F )| ≥ 3 and chromatic number q := χ(F ) ≥ 2.
We begin with two general constructions, providing lower bounds on Ts(n, F ).
Let us assume that n is a multiple of q − 1; regarding lower bounds for other orders we refer to the simple fact that This will give a fairly good approximation because the difference between the numbers of edges for two consecutive multiples of q − 1 will be O(n) only, while Ts will be shown to grow with a quadratic function of n. Even in a more general setting where F is not a fixed graph and q varies, say F = K ⌊ √ n⌋ , the difference between the numbers of edges will grow with O(qn) = o(n 2 ), which is negligible compared to Ts(n, F ).
The higher structure of both constructions is a partition of the n-element vertex set into p − 1 classes V 1 , V 2 , . . . , V p−1 , where • each |V i | is a multiple of q − 1, and • each V i induces a Turán graph for K q , i.e., the subgraph induced by V i in the graph of order n under construction is a complete multipartite graph with q − 1 vertex classes U i,1 , U i,2 , . . . , U i,q−1 of equal size, each U i,j is an independent set, and any two of them are completely adjacent.
In both constructions the degree classes will be V 1 , V 2 , . . . , V p−1 .
3 If p = 2, then either F = 2K 1 which is a singular subgraph of every graph with at least two vertices -hence Ts(n, F ) is meaningless -or F = K 2 and Ts(n, F ) = 0 for all n. The situation is similar if χ(F ) = 1, i.e. F = pK 1 , in which case Ts(n, F ) cannot be defined for n > (p − 1) 2 .
Construction 37. Choose the sizes of the degree classes V i in such a way that |V 1 | < |V 2 | < · · · < |V p−1 | holds, and under this condition |V 1 | is as large as possible, whereas |V p−1 | is as small as possible.
Since the sequence |V 1 |, |V 2 |, . . . , |V q−1 | is strictly increasing, we must have |U i,j | ≥ |U 1,j | + i − 1 for all 1 ≤ i < p − 1 (and all 1 ≤ j ≤ q − 1). Then the requirement on |V 1 | and |V p−1 | means that we need to maximize |U 1,1 | subject to By construction we also have: This implies that the degree sets are indeed the classes V i , and two types of singular subgraphs can occur: • subgraphs of a V i , thus having chromatic number less than q; • subgraphs with at most one vertex in each V i , thus having order less than p.
It follows that the constructed graph does not contain any singular subgraph isomorphic to F . Let us compare the number of edges with that in the Turán graph for K (p−1)(q−1)+1 .
Proof. Suppose first that n is divisible by q − 1. From the graph obtained in Construction 37 we can obtain the Turán graph if, for every 1 ≤ i ≤ p−1 2 and every 1 ≤ j ≤ q − 1, we replace the vertex classes U i,j and U p−i,j with two classes (independent sets) of sizes Due to the identity (x − a)(x + a) = x 2 − a 2 , this operation increases the number of edges proportionally to (p/2 − i) 2 , because the subgraph induced by U i,j ∪ U p−i,j remains a complete bipartite graph on exactly the same set of vertices and with an unchanged number of edges to its exterior. There are q − 1 choices for j, and i runs from 1 to ⌊(p − 1)/2⌋, hence the total difference grows with the order of qp 3 . If n = t(q − 1) + r with r = 0, then we supplement the construction with r isolated vertices, hence no singular F will arise while the number of edges does not decrease (actually remains unchanged). On the other hand, the Turán function clearly satisfies the inequality ex(n, H) − ex(n − r, H) < rn for every graph H and all natural numbers n and r.
Construction 39. For the sake of simpler description assume that n is a multiple of (p − 1)(q − 1), with q ≥ 2 and p ≥ 4. We define all U i,j to have the same size (1 ≤ i ≤ p − 1, 1 ≤ j ≤ q − 1), i.e. |U i,j | = n (p−1)(q−1) , each of them being an independent set; hence in particular Start with complete bipartite graphs between any two U i 1 ,j 1 , U i 2 ,j 2 . Represent the sets V i with single vertices v i , and view them as the vertices of K p−1 . It was proved by Chartrand et al. [13] that the edges of K p−1 can be assigned with integer weights from {1, 2, 3} in such a way that the weighted degrees of the vertices become mutually distinct. Now, for each vertex pair, • if the weight of v i 1 v i 2 is 1, keep complete adjacency between V i 1 and V i 2 ; • if the weight of v i 1 v i 2 is 2, remove a perfect matching between V i 1 and V i 2 ; • if the weight of v i 1 v i 2 is 3, remove a 2-factor between V i 1 and V i 2 .
By construction, the degree classes are the sets V i , hence the graph does not contain any singular subgraph with p vertices and chromatic number q; and the number of removed edges is at most (p − 2)n. In fact, applying the results of [21], this upper bound can be reduced to (p/2 + c)n, where c is a universal constant for all n and p, which is tight apart from the actual value of c.
Next, we prove an upper bound which shows that the constructions above give tight asymptotics on Ts(n, F ) for every fixed graph F as n gets large.
Moreover, for the complete graph K p (i.e., q = p) we have Both upper bounds are asymptotically sharp as n → ∞.
Proof. We begin with the inequality for K p , as it is much simpler to prove. If a graph G of order n has more than ex(n, K (p−1) 2 +1 ) edges, then by definition it contains a complete subgraph on (p − 1) 2 + 1 vertices. Among them, p have the same degree in G or p have mutually distinct degrees. Thus, G contains K p as a singular subgraph, which implies that Ts(n, K p ) cannot be that large.
In the general case let us asssume that G is a graph of order n, having as many as ex(n, K (p−1)(q−1)+1 ) + ǫn 2 edges. We are going to apply the Erdős-Stone theorem [18], which states that for any fixed ǫ > 0 a graph with n vertices and ex(n, K s ) + ǫn 2 edges contains not only a K s but also a complete multipartite graph with s vertex classes with t vertices in each class; here t can be taken any large as n increases. 4 We take s = (p − 1)(q − 1) + 1 and assume that n is large enough to ensure that also t is sufficiently large, say t ≥ p 2 .
Let A 1 , . . . , A (p−1)(q−1)+1 be the vertex classes of a K t,...,t ⊂ G. Each A i contains a singular B i ⊂ A i with |B i | ≥ √ t ≥ p, with vertices whose degrees are all equal or all distinct in G.
If the degrees are all distinct in at least p of the sets B i , then we can sequentially select one vertex from each B i such that in each step the degree of the selected vertex is distinct from all previously selected ones. This yields a singular K p ⊂ G, thus also F occurs as a singular subgraph of G.
Suppose that there are only h sets B i (where 0 ≤ h ≤ p − 1) inside which the degrees are distinct. We assume that these classes are the ones with largest subscripts, namely B (p−1)(q−1)−h+2 , . . . , B (p−1)(q−1)+1 . Then we obtain a sequence d 1 , d 2 , . . . , d (p−1)(q−1)−h+1 where d i is the degree of all vertices in B i . If this sequence contains q equal terms, then from the corresponding sets we can select the vertices for the color classes of F in a proper q-coloring, thus F is a singular subgraph of G with all degrees equal. Else every value occurs at most q − 1 times, hence the sequence contains at least z := (p−1)(q−1)−h+1 q−1 mutually distinct terms, which can be supplemented at least with further min(h, p − z) distinct degrees from the last h sets B i . Now we have with equality only if h = 0 or h = 1 or q = h = 2. Consequently, K p occurs as a singular subgraph of G with all degrees distinct.
Asymptotic tightness follows from the constructions described above, for both cases.
For the case p = q = 3 and n ≡ 2 (mod 4) we obtained an exact result.
We close this section with some fairly tight estimates for K 3 .
Proposition 42. For F = K 3 and n ≥ 3 we have the following inequalities. (iii) If n ≡ 3 (mod 4), then 3 8 n 2 − 1 4 n − 13 8 ≤ Ts(n, K 3 ) ≤ 3 8 n 2 − 11 8 . Proof. In all cases, the claimed upper bound is a Turán number minus 1, namely ex(n, K 5 ) − 1. Its validity follows from Theorem 40 by the further observation that the corresponding Turán graphs are unique and each of them contains a singular K 3 . For the lower bounds we give constructions as follows.
(i) This is a particular case of Construction 37, with |U 1,1 | = |U 1,2 | = 1 4 n − 1 and |U 2,1 | = |U 2,2 | = 1 4 n + 1. (ii) Start with the complete 4-partite graph with equal vertex classes of size 1 4 (n − 1), and join a new vertex, say z, to all vertices of two classes. Denoting n = 4h + 1, the two classes adjacent to z have degree 3h + 1, the other two classes have degree 3h, and z has degree 2h. There is no singular K 3 because there are only three distinct degrees (degree 2h occurring only on z), each degree class is triangle-free, and in every triangle containing z the other two vertices have degree 3h + 1.
(iii) Assume that n = 4h + 3. Start with the optimal construction for n − 1, that is the complete 4-partite graph with vertex classes of respective sizes h, h, h + 1, h + 1. Similarly to the case of (ii) join a new vertex z to the 2h vertices of the two smaller classes. Then 2h vertices have degree 3h + 3, 2h + 2 vertices have degree 3h + 1, and z has degree 2h alone, with all its neighbors having degree 3h + 3.
The principle of these constructions can also be applied to obtain improvements of the general lower bounds on Ts(n, F ) given in Proposition 38, for those n which are not divisible by q − 1.

Concluding remarks and open problems
There are several interesting directions deserving further study, which we only indicate briefly here. In fact some of the preceding results can be directly extended in one way or another, but a more systematic study would be necessary beyond pure generalizations.
The quadratic bound. We have seen that (R(G) − 1) 2 + 1 is an easy upper bound on Rs(G). On the other hand, from the graphs studied here it seems that this naive bound is not very bad. In this direction we propose the following conjecture.
(i) (weak form) There exists a constant c > 0 such that Rs(G) ≥ c (R(G)) 2 holds for all graphs G.
Remark 44. Proposition 17 implies the validity of part (i) for graphs containing very many isolated vertices. On the other hand the same proposition indicates that part (ii) needs the exclusion of isolates -or at least some related condition -because otherwise Rs(G n ) is quadratic in |V (G n )| rather than in R(G n ).
More than two colors. Instead of 2-coloring the edges of K m one may consider t ≥ 3 colors. In this case the notion of singular subgraph may be introduced in several ways; here we mention those two of them which can be considered weakest and strongest. In both of them we assume that t graphs G 1 , . . . , G t have been specified; moreover in any edge t-coloring of K m we consider the graphs F 1 , . . . , F t where the edge set of F j consists of the edges colored j. Let us introduce the following notions.
• A monochromatic subgraph H of color i is weakly singular if V (H) is singular in F i .
• A monochromatic subhraph H is strongly singular if V (H) is singular in F j for all 1 ≤ j ≤ t.
Then the weak singular Ramsey number Rs w (G 1 , . . . , G t ) is the smallest integer n such that, for every m ≥ n, every edge t-coloring of K m contains a weakly singular subgraph G i in the color class i for some 1 ≤ i ≤ t; and the strong singular Ramsey number Rs s (G 1 , . . . , G t ) is defined analogously. It can be proved in various ways that Rs w and Rs s are finite whenever the graphs G i are finite. As t grows, there is an increasing number of possibilities Isolated vertices. We have shown in Proposition 17 that the quadratic lower bound is tight whenever the number of non-isolated vertices is rather small compared to the order of the graph. Motivated by this, the following problem is of interest.
Problem 47. Given a graph G, determine the minimum number of isolated vertices which should be added to G so that the obtained graph G + satisfies the equality Rs(G + ) = (|V (G + )| − 1) 2 + 1.
Other structures. Ramsey theory has been studied for various structures, and the notion of singularity can be extended in a meaningful way in some of them. For example, for any family F of hypergraphs and for every natural number k, the inequality Rs(F , k) ≤ k(R(F ) − 1) 2 + 1 of Theorem 11 remains valid.
Singular Turán numbers. We have determined tight asymptotics for Ts(n, F ) for all graphs F having at least one edge, but the exact value is known in a small number of cases only. This leaves several interesting problems open.
Let us note that the upper bounds in Proposition 42 are tight for n = 4 and n = 5, while it seems plausible to guess that for every other n divisible by 4 the lower bound of (i) gives the correct value. In the other cases the lower bounds may turn out to be tight, at least asymptotically.
Problem 52. Determine the value of the constants c(F, r) for particular classes of graphs F , including complete graphs, complete bipartite graphs, paths and cycles.
Problem 53. Given a constant c in the range 0 < c < 1, find tight asymptotics on Ts(n, K cn ).
Problem 54. Given F , find tight asymptotics on the k-singular Turán numbers Ts(n, F, k).