On the minimum degree of minimal Ramsey graphs for multiple colours
A graph is -Ramsey for a graph , denoted by , if every -colouring of the edges of contains a monochromatic copy of . The graph is called -Ramsey-minimal for if it is -Ramsey for but no proper subgraph of possesses this property. Let denote the smallest minimum degree of over all graphs that are -Ramsey-minimal for . The study of the parameter was initiated by Burr, Erdős, and Lovász in 1976 when they showed that for the clique . In this paper, we study the dependency of on and show that, under the condition that is constant, . We also give an upper bound on which is polynomial in both and , and we determine up to a factor of .
A graph is -Ramsey for a graph , denoted by , if every -colouring of the edges of contains a monochromatic copy of . The fact that, for any number of colours and every graph , there exists a graph such that is a consequence of Ramsey’s theorem . Many interesting questions arise when we consider graphs which are minimal with respect to . A graph is -Ramsey-minimal for (or -minimal for ) if , but for any proper subgraph . Let denote the family of all graphs that are -Ramsey-minimal with respect to . Ramsey’s theorem implies that is non-empty for all integers and all finite graphs . However, for general , it is still widely open to classify the graphs in , or even to prove that these graphs have certain properties.
Of particular interest is , the complete graph on vertices, and a fundamental problem is to estimate various parameters of graphs , that is, of -Ramsey-minimal graphs for the clique on vertices. The best-studied such parameter is the Ramsey number , the smallest number of vertices of any graph in . Estimating , or even , is one of the main open problems in Ramsey theory. Classical results of Erdős  and Erdős and Szekeres  show that . While there have been several improvements on these bounds (see for example  and ), the constant factors in the above exponents remain the same. For multiple colours, the gap between the bounds is larger. Even for the triangle , the best known upper bound on the -colour Ramsey number is of order , whereas, from the other side, is the best known lower bound (see  for the best known constant).
Other properties of have also been studied: Rödl and Siggers showed in  that, for all and , there exists a constant such that, for large enough, there are at least non-isomorphic graphs on at most vertices that are -Ramsey-minimal for the clique . In particular, is infinite. Another well-studied parameter is the size Ramsey number of a graph , which is the minimum number of edges of a graph in .
Interestingly, some extremal parameters of graphs in could be determined exactly when the number of colours is two. In this paper, we consider the minimal minimum degree of -Ramsey-minimal graphs , defined by
where denotes the minimum degree of .
It is rather simple to see that, for any graph ,
Indeed, for , the proof of the lower bound is included in ; it generalises easily to more colours. We include a similar argument at the beginning of Section 3. In , Burr, Erdős, and Lovász showed that, rather surprisingly, the simple upper bound above is far from optimal when , namely .
In this paper, we study the behaviour of as a function of and . We mainly study as a function of with fixed. In particular, we determine up to a logarithmic factor.
There exist constants such that for all , we have
One can show that (this follows from a stronger statement, cf. Theorem 1.5 and Proposition 3.2). However, it is not clear that . Therefore, the lower bound on does not necessarily imply a similar lower bound on . We can in fact only prove a super-quadratic lower bound on that is slightly weaker.
For all there exist constants such that, for all ,
The proof of the upper bounds in Theorems 1.1 and 1.2 are of asymptotic nature and require to be rather large. Moreover, the exponent of the -factor in the latter upper bound depends on the size of the clique. Therefore, we also prove an upper bound on which is polynomial both in and in and is applicable for small values of and .
For , ,
Tools. We give an overview of the tools we use to prove bounds on . The first step will be to reduce finding to a simpler problem. We call a sequence of pairwise edge-disjoint graphs on the same vertex set a colour pattern on . For a graph , a colour pattern is called -free if none of the contains as a subgraph. A graph with coloured vertices and edges is called strongly monochromatic if all its vertices and edges have the same colour.
The -colour -clique packing number, , is the smallest integer such that there exists a -free colour pattern on an -element vertex set with the property that any -colouring of contains a strongly monochromatic .
While Burr, Erdős, and Lovász  do not explicitly define in their proof of , they do essentially show that and it is then not hard to see that . Here we generalise their result to an arbitrary number of colours.
For all integers we have .
The lower bound is not difficult to derive from the definitions. The upper bound follows from a powerful theorem of . We use later generalisations of this theorem by Burr, Nešetřil, and Rödl  and, recently in 2008, by Rödl and Siggers  to derive for arbitrary .
The problem then becomes to obtain bounds on . We will see that relates closely to the so-called Erdős-Rogers function, which was first studied by Erdős and Rogers  in 1962. We will be particularly concerned with the special case of the Erdős-Rogers function, denoted by , which is defined to be the largest integer so that in any -free graph on vertices, there must be a vertex-set of size that contains no . For our bounds, we will rely heavily on the modern analysis of found in [9, 10, 11, 21]. In Section 3, we will see that essentially , so lower bounds on directly translate to lower bounds on . In Section 4, we obtain upper bounds on by packing graphs, each giving good upper bounds on , into the same vertex set.
2 Passing to
For all , we have .
Let be an -Ramsey-minimal graph for with a vertex of degree . Let be an -colouring of without a monochromatic ; such a colouring exists by the minimality of . Let be the pairwise edge-disjoint subgraphs of the colours within the neighbourhood of ; they form a -free colour pattern on . We show that any vertex-colouring of must contain a strongly monochromatic -clique and hence, by the definition of , the number of vertices must be at least . Indeed, given any vertex-colouring of we may define an extension of to the edges incident to by colouring an edge with the colour of the vertex . Since is -Ramsey for , this extension of contains a monochromatic -clique . Moreover, must contain (as was free of monochromatic ). By the definition of the extension of , the vertices of in form a strongly monochromatic in . ∎
In order to show , we first prove a theorem that guarantees, for any integer and graph which is -connected or a triangle, a fixed colour pattern on a given induced subgraph of some graph which is not -Ramsey for , in any monochromatic -free -colouring of . A similar theorem was proved for and for in , where they use it to show . The tools used to prove this were generalised to any -connected graph in , and, more recently, to any number of colours and any graph which is -connected or a triangle .
Let be any -connected graph or and let be an -free colour pattern. Then there is a graph with an induced copy of the edge-disjoint union so that and in any monochromatic -free -colouring of each is monochromatic and no two distinct and are monochromatic of the same colour.
We use the idea of signal sender graphs which was first introduced by Burr, Erdős and Lovász . Let and be integers and be a graph. A negative (positive) signal sender () is a graph with two distinguished edges of distance at least , such that
in every -colouring of without a monochromatic copy of , the edges and have different (the same) colours.
We call and the signal edges of .
Burr, Erdős and Lovász  showed that positive and negative signal senders exist for arbitrary in the special case when the number of colours is two, and is a clique on at least three vertices. Later, Burr, Nešetřil and Rödl  extended these results to arbitrary -connected . Finally, Rödl and Siggers  constructed positive and negative signal senders and for any , as long as is -connected or .
Let be a graph that is either -connected or and let be an -free colour pattern on vertex set . We construct our graph using the signal senders of Rödl and Siggers. We first take the graph on which is the edge-disjoint union of the edge sets of the graphs and add isolated edges disjoint from . Then for every and every edge we add a copy of , such that and are the two signal edges and the sender graph is otherwise disjoint from the rest of the construction. Finally, for every pair of edges , we add a copy of , such that and are the two signal edges and the sender graph is otherwise disjoint from the rest of the construction.
By the properties of positive and negative signal senders, in any -colouring of without a monochromatic , each must be monochromatic and no two may be monochromatic in the same colour.
Now we need only to show that there exists an -colouring of with no monochromatic . For this, we first colour each with colour . Then, we extend this colouring to a colouring of each signal sender so that each signal sender contains no monochromatic copy of . This is possible since each positive (negative) signal sender has a colouring without a monochromatic copy of in which the signal edges have the same (different) colours. Let us consider a copy of in . We will see that is contained either within or within one of the signal senders and hence it is not monochromatic. If this was not the case, then there would be a vertex of that is not in any of the signal edges, that is, for some signal sender but not contained in any of the two signal edges of . Since is not entirely in , there must be a vertex . This immediately implies that , since and are not adjacent. Since is -connected there are three internally disjoint -paths in . These paths can leave only through one of its two signal edges. Hence there is a path of in between the two signal edges. This is a contradiction because the distance of the two signal edges in is at least . ∎
Let a -free colour pattern be given on vertex set with , so that any -colouring of contains a strongly monochromatic . Take as in Theorem 2.2 with , and define to be with a new vertex which is incident only to . We claim that , that is for any -colouring of we find a monochromatic . If already the restriction of to contains a monochromatic then we are done. Otherwise, by Theorem 2.2, we have that, after potentially permuting the colours, each subgraph is monochromatic in colour . We define a colouring of by colouring with . Then, by the choice of , there is a strongly monochromatic clique in . This clique along with vertex forms a monochromatic in the colouring .
So . Now observe that any -Ramsey-minimal subgraph of must contain the vertex , since is not -Ramsey for by Theorem 2.2. Hence for the minimum degree of any -Ramsey-minimal subgraph we have that . ∎
3 Lower bounds on
First, we prove a simple linear lower bound on . This simple estimate will later be used to obtain a super-quadratic lower bound.
For all and , we have .
We will show that for any given colour pattern on vertex set , , there is a vertex-colouring of without a strongly monochromatic and hence, . Observe that every vertex has degree at most in at least one of the colour classes, say . Colouring vertex with colour ensures that is not contained in any strongly monochromatic , as its degree in is too low. Hence, as promised, this vertex-colouring of produces no strongly monochromatic . ∎
For a graph , the -independence number is the largest cardinality of a subset without a . For , this is the usual independence number . Recall that the Erdős-Rogers function is defined to be the minimum value of over all -free graphs on vertices.
The following proposition provides the recursion for our lower bound.
For all we have that satisfies the following inequality:
Take to be a -free colour pattern on vertex set , , so that any -colouring of the vertices contains a strongly monochromatic . Let be a -independent set of size in the graph . We claim that the -free colour pattern restricted to the vertex set has the property that any -colouring contains a strongly monochromatic . Indeed, the extension of to which colours the vertices in with colour must contain a strongly monochromatic and this must be inside , since does not contain at all. Hence and then, since is a -free graph on vertices, we have that
Therefore, we are interested in good lower bounds on the Erdős-Rogers function . It is easy to see that every -free graph on vertices contains a -free set of size at least . If there exists a vertex of degree at least , then is a -free set of size at least . Otherwise, and we can use the well-known fact that (cf. ) to deduce that . Therefore, .
A result of Shearer  implies that , which is the best known lower bound on . Bollobás and Hind  proved that . This lower bound was subsequently improved by Krivelevich . Recently, Dudek and Mubayi  showed that this result can be strengthened to
by using a result of Shearer .
Proof of the lower bounds in Theorem 1.1 and 1.2.
Let be fixed and and for brevity let us write .
Let for , where
is a non-decreasing function such that
for with some constant .
Note that one can take by  and for
one can take with some constant by .
We show that there exists a constant such that for ,
We prove this statement by induction on . For this is true provided is chosen small enough. For , by Proposition 3.2 and since is non-decreasing, we have that
Using the induction hypothesis, Lemma 3.1 and that is non-decreasing for , we obtain
By our assumption on the last term is positive, provided is small enough. ∎
4 Packing -critical graphs
In this section we prove the upper bounds in Theorems 1.1, 1.2 and 1.3. Our task is to derive upper bounds for , that is we want to find -free colour patterns such that every -colouring of the vertices produces a strongly monochromatic . Let us first motivate the idea behind our proofs. Given a colour pattern on an -element vertex set and any -colouring of , at least one of the colours, say , occurs times. If every set of at least vertices in contains a , then we must have a strongly monochromatic clique in colour . This motivates the following definition: we call a graph on vertices -critical if and . We have thus obtained the following lemma.
If there exists a colour pattern where each is -critical, then .
For the rest of this section, we will focus on packing edge-disjoint -critical graphs into the same -element vertex set, such that is as small as possible.
In order to produce at least one -critical graph, let us recall the Erdős-Rogers function, defined as , where the minimum is taken over all -free graphs on vertices. By definition, we have for all that
So the question whether at least one -critical graph exists on vertices is equivalent to the question whether .
When , an -critical graph is precisely an -vertex triangle-free graph with independence number less than . Hence an -critical graph exists if and only if . It is known that where the upper bound was first shown by Ajtai, Komlós and Szemerédi  and the matching lower bound was first established by Kim . Therefore, if is an -critical graph, then for some constant , and -critical graphs do exist for for some constant . For our purpose, however, we need to pack many -critical graphs in an edge-disjoint fashion into vertices. The next lemma states that we can do so at the expense of a factor of .
Let be an integer. Then there exists a colour pattern on vertex set , where , such that each is -critical.
For fixed , Dudek, Retter, and Rödl  recently showed that . That is, they constructed a -free graph on vertices (where is large enough) such that every subset of vertices contains a . This is an -critical graph with . Again, we would like to pack of those graphs into . But rather than taking a fixed -critical graph and pack it into , we construct (edge-disjoint) -critical graphs simultaneously as subgraphs of . As it turns out, this simultaneous construction is only little harder than the construction itself in ; we prove it by black-boxing theorems from .
For all integers there exist a constant and such that, for all , the following holds. There exists a colour pattern on vertex set , where , such that each is -critical.
For the upper bound in Theorem 1.3, we are motivated by graphs constructed by Dudek and Rödl in . The graph on vertices constructed in  is -critical with . Here it is not as clear to just refer to lemmas from  in order to do a “simultaneous” construction. So we will start the construction from scratch and provide all the details needed.
Let . Then there exists a colour pattern on vertex set , where , such that each is -critical.
4.1 Proofs of the Lemmas
Packing many -free graphs with small independence number.
Here, we prove Lemma 4.2. To that end, we will show the existence of a graph on vertices, where , which can be written as a union of edge-disjoint graphs , …, which are all -free and without independent sets of size . We will find the graphs successively as subgraphs of using the following.
Lemma 4.5 (Lovász Local Lemma, see, e.g., [2, Lemma 5.1.1]).
Let be events in an arbitrary probability space. A directed graph on the set of vertices is called a dependency digraph for the events if for each , , the event is mutually independent of all the events . Suppose that is a dependency digraph for the above events and suppose there are real numbers such that and for all . Then
In particular, with positive probability no event holds.
Given , set and . For a graph on vertices, we define () to be the smallest (largest) number of edges that appear in any subset of size . The following lemma is the crucial step to find the graphs .
Let be a graph on vertices, where is large enough, and assume . Then there is a subgraph on the same vertex set such that is triangle-free, has no independent set on vertices, and .
Let and . Choose by including each edge of independently with probability . For a subset , let and denote the number of edges in and , respectively. It suffices to show that is triangle-free, , and with positive probability. To that end, we want to apply the Lovász Local Lemma, and, therefore, we define the set of bad events in the natural way. Namely, for every that forms a triangle in , we set to be the event that is a triangle as well. Clearly, the probability of such an event is . Further, for every , we set to be the event that either is an independent set in or satisfies . Then,
Note that and , since , so that for large enough
Let be the collection of bad events. That is, . In the auxiliary dependency graph , we connect two of the events if . Then is mutually independent from the family of all for which is not an edge in this dependency graph. To apply the Lovász Local Lemma, we now bound the degrees in . We denote by the neighbours in the dependency graph of the event . If we have
|If we have|
Therefore, by Lemma 4.5, if there exist real numbers such that
then there exists a graph such that none of the events in occurs. We show that these two conditions are fulfilled for and . First note that, for large enough,
We use and for to claim (4) holds if
Now, and . So (4) holds if
which is satisfied by choice of . Applying Lemma 4.5 yields the existence of a subgraph such that none of the events in hold, i.e. has the desired properties. ∎
Proof of Lemma 4.2.
Let large enough be given, and set and as before. Define . We choose our graphs inductively as subgraphs of ; given for such that , we have since that
so, by Lemma 4.6, we may find a subgraph of with such that is triangle-free and has no independent set on vertices. Then take . The graph will be edge-disjoint from (and, inductively, from ), and
as desired. ∎
Proof of Lemma 4.3.
Fix and let be large enough. We need to construct graphs on vertices that are -free, but every subset of size contains a . Let be the largest prime power such that