On the Profile of Multiplicities of Complete Subgraphs

On the Profile of Multiplicities of Complete Subgraphs

Uriel Feige     Anne Kenyon     Shimon Kogan

Department of Computer Science and Applied Mathematics
Weizmann Institue, Rehovot 76100, Israel

uriel.feige@weizmann.ac.il
annemkenyon@gmail.com
shimon.kogan@weizmann.ac.il
Abstract

Let be a -coloring of a complete graph on vertices, for sufficiently large . We prove that contains at least monochromatic complete subgraphs of size , where

The previously known lower bound on the total number of monochromatic complete subgraphs, due to Székely [Szé84], was . We also prove that contains at least monochromatic complete subgraphs of size .

If furthermore one assumes that the largest monochromatic complete subgraph in is of size (it is a well known open question whether such graphs exist), then for every constant we determine (up to low order terms) the number of monochromatic complete subgraphs of size . We do so by proving a lower bound that matches (up to low order terms) a previous upper bound of Székely [Szé84]. For example, the number of monochromatic complete subgraphs of size is .

1 Introduction

The “classic” diagonal Ramsey number question asks the following: what is the minimum such that all -colored (edge colored) complete graphs of size contain a monochromatic complete subgraph of size ? Letting be the size of the largest monochromatic complete subgraph, the diagonal Ramsey number question can be rephrased as: what is the minimum over all possible -colorings of the complete graph on vertices?

These questions are the basis for the field of Ramsey Theory. Known bounds say that all graphs of size contain a monochromatic complete subgraph of size at least and that there are graphs with a maximum monochromatic complete subgraph of size at most . These results date back to 1935 (Erdös and Szekeres [ES35]) and 1947 (Erdös [Erd47]), respectively. There have since been improvements to these bounds, but only to the lower order terms.

In this paper we turn our attention to a related question that is referred to as “Ramsey Multiplicity”: what is the minimum number of monochromatic complete subgraphs of size in a -colored complete graph? The classic Ramsey problem is a special case of the Ramsey multiplicity question, in the sense that it asks to determine the largest value (as a function of ) for which the Ramsey multiplicity is guaranteed to be nonzero. Hence beyond its intrinsic interest, progress on the Ramsey Multiplicity question may potentially lead to progress on the classic Ramsey problem. Our interest in this work is mainly in the case that , which is the relevant range for the classic Ramsey problem. There has also been previous work for the case of constant (see the Section 1.4 for more details).

1.1 Main results

The question of the total number of monochromatic complete subgraphs is addressed by Szekely [Szé84], who shows that for large enough , in any -coloring of a complete graph on vertices there are at least monochromatic complete subgraphs. We improve this result by proving the following in Section 3.

Theorem 1.1.

Let be a -coloring of a complete graph on vertices. Then for any large enough , graph contains at least monochromatic complete subgraphs of size where

Equivalently, every 2-coloring of a complete graph on vertices gives at least roughly monochromatic subgraphs of size in the range .

Random graphs provide the known upper bound on the number of monochromatic complete subgraphs (see [Szé84] for details).

Theorem 1.2.

For all large enough , there is a -coloring of a complete graph on vertices with at most monochromatic complete subgraphs.

As noted above, it is a long standing open question whether for every there are 2-colorings of the complete graph of size that do not induce a monochromatic complete subgraph of size . Not wishing to carry in our notation, we refer to the upper bound on the size of monochromatic complete subgraphs as . Adapting previous terminology by which a 2-colored complete graph of size with no monochromatic complete subgraph of size is referred to as -Ramsey, we say that a 2-colored complete graph with vertices is Half-Ramsey if it does not contain a monochromatic complete subgraph of size .

It is not known whether Half-Ramsey graphs exist at all. Here we assume that such graphs do exist, and then study what the property of being Half-Ramsey implies about questions concerning Ramsey multiplicities. One may hope that these implications will either help in actually exhibiting Half-Ramsey graphs, or result in a contradiction that will show that there are no Half-Ramsey graphs.

In Section 4.1 we review known upper bounds on the number of monochromatic complete subgraphs of a Half-Ramsey graph. These upper bounds were derived by Szekely [Szé84]. One such bound is the following.

Theorem 1.3.

Let be a Half-Ramsey graph on vertices. Then has at most

monochromatic complete subgraphs.

In Section 4.2 we provide a matching lower bound, up to low order terms.

Theorem 1.4.

Let be a Half-Ramsey graph on vertices. Then has at least

monochromatic complete subgraphs.

Moreover, for Half-Ramsey graphs we do not only determine the total number of complete subgraphs, but also determine what we refer to as the profile of Ramsey multiplicities. Namely, for every we determine (up to low order terms) the number of monochromatic complete subgraphs of size . This number is for some concave monotonically increasing function that is defined in Section 4.3. See more details in Section 4.3.

1.2 Additional results

We have some results that shed light on the profile of multiplicities of arbitrary graphs (without requiring them to be Half Ramsey).

In Section 3 we prove a bound that holds for all .

Theorem 1.5.

For all , any -coloring of the edges of the complete graph on vertices contains at least

monochromatic complete subgraphs of size .

Furthermore we prove in Section 3 the following variant of Theorem 1.1.

Theorem 1.6.

For every there is a constant such that the following statement holds.
Given a natural number and a natural number . For every -coloring of the edges of the complete graph on vertices, there is a (which depends on ) such that contains at least

monochromatic complete subgraphs of size .

In Section 5 we provide improved bounds for sufficiently large .

Theorem 1.7.

Let be a -coloring of a complete graph on vertices. Then for any large enough even , graph contains at least monochromatic complete subgraph of size .

Furthermore we prove in Section 5 a bound on the number of monochromatic complete subgraphs of size at most .

Theorem 1.8.

Let be a -coloring of a complete graph on vertices. Then for any large enough , graph contains at least monochromatic complete subgraphs of size at most , where .

Another question that we address is the average size of a monochromatic complete subgraphs in a -coloring of a complete graph. We prove the following in Section 6.

Theorem 1.9.

Let be a -coloring of a complete graph on vertices. Then for any large enough , the average size of a monochromatic complete subgraph in is at least

The following upper bound can be easily derived by considering random graphs.

Theorem 1.10.

For all large enough , there is a -colorings of a complete graph on vertices, in which the average size of a monochromatic complete subgraph is at most .

Recall that we show that the profile of multiplicities of Half Ramsey graphs has the property that there is a large number of monochromatic complete subgraphs of size roughly (in fact, almost all monochromatic complete subgraphs are of this size), but there is no monochromatic complete subgraph of slightly larger size . As noted above, we do not know if Half Ramsey graphs exist, and hence it is natural to ask whether one can exhibit graphs for which the profile of multiplicities has qualitatively similar properties (e.g., a sudden drop to 0 in the number of monochromatic complete subgraphs). This is one of the motivations for Section 7 that discusses relationships between the maximum size, the average size and the total number of monochromatic complete subgraphs. Among other results, we show:

Theorem 1.11.

There is a graph of size in which the average size of a monochromatic complete subgraph satisfies , and there is no monochromatic complete subgraph of size .

1.3 Some Notes, Definitions, and Background

In this paper will denote the binary logarithm, while will denote the natural logarithm.
Some of the related and cited work talk about cliques and independent sets, which are equivalent to a monochromatic complete subgraph in a -colored complete graph, if we let one color be edges and the other color be non-edges. Therefore, when we discuss or use their results we sometimes re-word them to coloring terminologies, without further comment.

Let the Ramsey Number be the minimum size such that all -colored graphs of this size have either a blue monochromatic complete subgraph of size or a red monochromatic complete subgraph of size . Ramsey’s Theorem states that there exists a positive integer such that this holds. We also state the Erdös-Szekeres bound [ES35].

Let be the diagonal Ramsey Number, when . Simple known bounds for the diagonal are of the form:

There exist improvements on these bounds in lower order terms, but we do not use them in this paper so we do not include them here.

Let be the number of monochromatic complete subgraphs of size in a graph of size . Let

Let , and lastly let , so that gives the minimum fraction of all subsets of size that are monochromatic complete subgraphs. This notation is consistent with the related work.

Let denote a clique of size .

1.4 Related Work

The work most related to this paper is that of Székely [Szé84]. He showed that for large enough , in any -coloring of a complete graph on vertices there are at least monochromatic complete subgraphs, and there is -coloring with at most monochromatic complete subgraphs. Our Theorem 1.1 improves over his lower bound. In the same work, Székely also provides upper bounds on Ramsey multiplicities for Half-Ramsey graphs. We prove lower bounds that match his upper bounds (up to low order terms). See Corollary 4.26.

The study of the multiplicity of monochromatic compete subgraphs was introduced by Erdös in 1962 in [Erd62], where Erdös proves that for all graphs,

(1.1)

In the same paper, Erdös proves using the probabilistic method that

(1.2)

Erdös conjectured that the upper bound in Inequality 1.2 is tight, or in other words, that an Erdös-Rényi random graph is the graph with the smallest number of monochromatic complete subgraphs of every size. In 1959, this conjecture was proved true for the case by Goodman [Goo59].

A survey on Ramsey Multiplicity results was published in 1980 by Burr and Rosta [BR80], in which they extend Erdös’s conjecture to the multiplicity of any subgraph, not just monochromatic complete subgraphs.

The conjecture was later disproved by counterexamples in 1989 by Thomason [Tho89], who showed that it does not hold for . Subsequently, several others worked on upper bounds for for small . Soon after Thomason’s work Franek and Rödl [FR93] also gave some different counterexamples based on Cayley graphs for . Then in 1994, Jagger, Št’ovíček and Thomason [JŠT96] studied for which subgraphs the Burr-Rosta conjecture holds, and found that it does not hold for any graph with as a subgraph, which is consistent with the result found by Thomason.

On the flip side, with regards to the lower bound in Inequality 1.1, in 1979 Giraud [Gir79] proved that . More recently in 2012, Conlon [Con12] proved that there must exist at least monochromatic complete subgraphs of size , in any -colouring of the edges of , where and is a constant independent of . This result is incomparable with our Theorem 1.6.

We also give bounds in this paper on the average size of a monochromatic complete subgraph in a graph . Furthermore we study the ratio between the average size of a monochromatic complete subgraph and the size of a maximum monochromatic complete subgraph in a graph . There appears to be no previous work directly on this topic. However, one of the primary motivations for our work on this topic is from the study of the minimum of the maximum independent set size over all -free graphs of size .

In 1995, Shearer [She95] used the probabilistic method to prove that , where is the size of the maximum independent set and is the average degree in the graph. Following his technique, Alon [Alo96] proved that for a graph in which the neighborhood of every vertex is -colorable, for some constant . Note that an -colorable graph is -free, since a clique can contain at most one vertex of each color.

The latest improvement for -free graphs is due to Bansal, Gupta and Guruganesh [BGG15], proving that . There is still a gap in this question, the upper bound being for -free graphs (also given in Bansal et al [BGG15]). All three of these papers actually prove that a random independent set in is of the given size, and then conclude that therefore the maximum independent set must be at least that size as well.

Thus, knowing the relationship between a random independent set and a maximum one could be useful in improving these bounds. Analogously we study in this paper the relationship between the average size of a monochromatic complete subgraph and the size of a maximum monochromatic complete subgraph.

2 The construction of Ramsey Trees

Let be the set of vertices of a complete graph on vertices. Let red and blue be the two colors of the edges of . Let be the set of vertices connected to vertex by a red edge, and be the set of vertices connected to by a blue edge. We shall describe several variants of a data structure which we call a Ramsey Tree. For the sake of clarity, we will use the term vertex to refer to vertices of a graph , and the term node to refer to nodes of the respective Ramsey tree. Estimates on the number of nodes in various levels of the Ramsey tree will allow us to obtain bounds on the number of complete subgraphs of (see Lemma 2.1 for example). The most general variant is the General Ramsey Tree (GRT). Other variants, the Biased Ramsey Tree (BRT) and the restricted Ramsey Tree (RRT), are subtrees of the GRT. They are introduced because their structure is more regular than that of the GRT, and this simplifies the derivation of various estimates that are used in our proofs.

2.1 Construction of a General Ramsey Tree

We shall describe how to build a General Ramsey Tree (GRT) from the graph . With each node in the tree, we will associate a vertex in the graph , the level of the node, and a bag , where a bag is simply a set of vertices that will be explained in a few lines. There will be many nodes in the tree construction associated with a given vertex .

We will build trees and later will connect them into one tree . Each tree is rooted at node , where so that we have one tree per vertex in . Each root node has bag . Furthermore we set the level of each root node to be .

Each tree is built recursively, as follows. There is one child of node for every vertex in the bag , and for each such child we set the level . The children of node are split into left and right children. The left children correspond to vertices in the set and the right children correspond to vertices in the set , where and are sets of vertices satisfying which we define in the following manner:

  • contains all the vertices in that are connected to in by a red edge, or in other words: . For each left child corresponding to vertex , we let its bag .

  • contains all the vertices in that are connected to in by a blue edge, or in other words: . For each right child corresponding to vertex , we let its bag .

We apply this recursively, beginning at the root of the tree, then the new children nodes of the root, then their children, et cetera. The recursions end when the bags are all empty, and then we have our completed tree .

We add a dummy node (which we denote as the super-root) at level and its bag will contain all the vertices of graph (that is the bag is of size ). The General Ramsey Tree (GRT) is obtained by connecting the super-root to the roots of the trees.

Now we shall describe some properties of the GRT . Let be the set of nodes on level of the GRT . The following two lemmas provide lower and upper bounds on the number of monochromatic complete subgraphs in , as a function of the sizes of (for various levels ).

Lemma 2.1.

If for some we have , then graph contains at least monochromatic complete subgraphs.

Proof.

The set of vertices corresponding to the nodes in a given path starting from a root node and ending in level of the GRT can appear in at most (l+1)! orders. Hence we have at least such paths where no two paths correspond to the same set of vertices in . Now consider one of these paths. Every edge in the path is either ”going left” or ”going right”. In other words, each edge either limits the new bag to a red-connected neighborhood or a blue-connected neighborhood of the parent node’s corresponding vertex in . Hence each path induces a red monochromatic complete subgraph in and a blue monochromatic complete subgraph in in the following manner: We can take one monochromatic complete subgraph to be vertices in corresponding to the parent nodes of ”left going” edges and the second monochromatic complete subgraph consist of vertices in corresponding to the parent nodes of ”right going” edges (if the last node in the path has no children we add its corresponding vertex arbitrarily to one of the monochromatic complete subgraphs). Now notice that no two such paths can induce the same two monochromatic complete subgraphs and , as every path corresponds to a different set of vertices in . Hence we have at least monochromatic complete subgraphs in and we are done.    

Lemma 2.2.

If for some we have , then graph contains at most monochromatic complete subgraphs of size .

Proof.

This follows from the fact that every permutation over the vertices of a monochromatic complete subgraph of size appears as a path starting at a root node and ending at level in the GRT .    

Lemma 2.3.

For all , if then .

Proof.

Notice that if then for all , as each node has a parent in the GRT . Let be the nodes on level of the GRT , thus we have . Furthermore by the definition of the GRT we have

(2.1)

and

(2.2)

where Inequality (2.1) follows from the fact that is minimized when .

Hence

(2.3)

where Inequality (2.1) follows from the Cauchy-Schwartz inequality. Thus we are done.    

Lemma 2.4.

For all , if then .

Proof.

if then for all , as each node has a parent in the GRT. Now we will prove the lemma by induction on . Notice that as we have root nodes in the GRT and furthermore as a bag associated with a root node is of size . Hence the base case of the induction follows as .
Now assume that the lemma holds for , we will prove that the lemma holds for . By Lemma 2.3 we have

and by the induction hypothesis we have

Hence we have

and we are done.    

Lemma 2.5.

If for some and , we have then for all for which .

Proof.

The proof is almost identical to Lemma 2.4 and thus omitted.    

2.2 Biased Ramsey Trees

We start by describing how to build a Biased Ramsey Tree (BRT) from the graph .

In this construction each node of the BRT will have a bias parameter such that . Furthermore we always assume that all nodes on the same level of the tree have the same bias (the definition of a level is given in the definition of the BRT below).

As before for GRT, with each node in the tree, we will associate a vertex in the graph , the level of the node, a bag , and the color of the node (which was implicit in GRTs). In addition, we also associate a bias with the node, and a parameter related to .

We will build trees (and later connected them into one tree). Each Tree is rooted at node , where so that we have one tree per vertex in . Each root node has bag which is defined as follows: If then we set and we set to be an arbitrary subset of of size , that is and , furthermore we set the color of the node to be red. Otherwise we set and we set to be an arbitrary subset of of size , that is and , furthermore we set the color of the node to be blue. We set the level of each root node to be .

Now we will explain how to build the trees. Each tree is built recursively, as follows. There is one child of node for every vertex in the bag and for each such child we set the level . We define for each child of node sets and in the following manner:

  • contains all the vertices in that are connected to in by a red edge, or in other words: .

  • contains all the vertices in that are connected to in by a blue edge, or in other words: .

Notice that . If then we set and we set the bag to be a subset of of size , that is and , furthermore we set the color of the node to be red. Otherwise we set and we set to be a subset of of size , that is and , furthermore we set the color of the node to be blue.

We apply this recursively, beginning at the root of the tree, then the new children nodes of the root, then their children, et cetera. The recursions end when the bags are all empty, and then we have our completed tree .

We add a dummy node (which we denote as the super-root) at level and its bag will contain all the vertices of graph (that is the bag is of size ). The Biased Ramsey Tree (BRT) consists of these trees, with roots connected to the super-root.

2.3 Ramsey Trees (of bias )

Biased Ramsey trees in which the bias of all the nodes in the tree is will be particularly convenient for us, especially when the number of vertices in the graph is a power of 2. Hence we shall reserve the term Ramsey Tree (without mentioning the bias explicitly) to refer to a Biased Ramsey Tree with bias , for a graph whose number of vertices is . The Ramsey Tree contains levels (not including level ), and the bags at level contain one vertex each. More generally, the bag size of nodes on level in the Ramsey Tree is .

We color the nodes in the final level of the Ramsey Tree in the following manner. Let be a node in level of the Ramsey Tree. Look at the path from level up to the parent of node (this parent is on level ). If this path contains at least red nodes we color node with red, that is we set to be red, otherwise we shall set to be blue. This coloring ensures us the following fact.

Lemma 2.6.

For any -colored complete graph on vertices, each path from a root node to a node in level in the corresponding Ramsey Tree contains at least nodes of the same color. Furthermore the vertices corresponding to the nodes of the same color in induce a monochromatic complete subgraph in .

Proof.

A path from a root node to a node in level in the Ramsey Tree contains nodes. Let be the last node in path and let be the parent of in the path . The path from ro contains nodes and thus it contains at least nodes of the same color, assume without loss of generality that this color is red. Thus by the definition of the Ramsey Tree node will also be colored red. We conclude that the path contains at least nodes of the same color.

Now we shall prove that the vertices corresponding to the nodes of the same color in induce a monochromatic complete subgraph in . Let be a set of nodes of the same color in , assume without loss of generality that this color is red. Then for any node the vertex is connected to all the vertices in by red edges. Hence the vertices corresponding to the nodes in induce a red monochromatic complete subgraph in .    

Lemma 2.7.

For any -colored complete graph on vertices , the corresponding Ramsey Tree contains exactly paths from a root node to a node on level .

Proof.

Follows from the fact that a bag on level in the Ramsey Tree is of size , hence the number of paths from a root node to a node on level is

 

2.4 Restricted Ramsey Trees

Now we describe the construction of Restricted Ramsey Tree (RRT) from the graph . A Restricted Ramsey Tree is an induced sub-tree of the Biased Ramsey Tree associated with graph , which is defined in the following manner. Let be the set of nodes on level of the BRT . If contains more red nodes than blue nodes then level of will contain all the red nodes of , otherwise level of will contain all the blue nodes of . Now let be all the nodes in level of which are children (in ) of nodes in level of . If contains more red nodes than blue nodes then level of will contain all the red nodes of , otherwise level of will contain all the blue nodes of . And we continue recursively: let be all the nodes in level of which are children (in ) of nodes in level of . If contains more red nodes than blue nodes then level of will contain all the red nodes of , otherwise level of will contain all the blue nodes of .

Denote the set of nodes in level of by . Recall that we will always assume that for each the bias of all the nodes in is the same and we will denote it by . Hence we have for all that . In particular we can set this value for level as . Furthermore by our construction of the RRT we have for each that the nodes in have the same color and we will denote this color by . Finally we denote by the size of the bags of the nodes in (all such nodes have the same bag size by the construction of the RRT).

Lemma 2.8.

Given a RRT , if there is a such that for all we have then we have for all the following.

Proof.

We will prove by induction on . The bast case follows from the fact that . Now we will assume that the claim holds for , and will prove it for . By the definition of the RRT we have

by the induction hypothesis

And thus we are done.    

Suppose that is the last level of the RRT (that is the bags of nodes on level are empty) and let . Recall that as the bag size of the super-root is .

Lemma 2.9.

Let be a set of level indices, where the nodes in all the levels in of the RRT are of the same color, that is for all we have . Then the graph associated with the RRT contains at least

monochromatic complete subgraphs of size .

Proof.

Assume . Fix a monochromatic complete subgraph of size . We shall denote a path from a root node to a node on level as a full path. There are at most

(2.4)

different full paths in which induce the monochromatic complete subgraph on the levels of , as there are orders in which can appear on the levels of . Now by the definition of the RRT the total number of full paths in is at least

(2.5)

We conclude that the number of monochromatic complete subgraphs in of size is at least

and the proof follows.    

3 Monochromatic Complete Subgraphs

One formulation of Ramsey’s theorem is the following

Theorem 3.1.

Any -coloring of the edges of the complete graph on vertices contains a monochromatic complete subgraph of size .

We will start by proving the following strengthening of Theorem 3.1.

Theorem 3.2.

For all , any -coloring of the edges of the complete graph on vertices contains at least

monochromatic complete subgraphs of size .

Proof.

Let be a -coloring of the edges of the complete graph on vertices. Let be the Ramsey Tree of as defined in Section 2.3. Notice that by Lemma 2.6 any path of length (that is a path from a root node to a node in level ) in contains either red nodes or blue nodes which correspond to a red or a blue monochromatic complete subgraph of size in . Henceforth we will denote a path of length in as a full path.

Let . Let where and (notice that and ). Given a monochromatic complete subgraph of size in , it can appear in at most

(3.1)

full paths in in which the vertices of correspond to nodes at levels

of the path. This follows since the sizes of the bags in which the vertices of appear are simply

and the product of the sizes of the remaining bags in a full path is . The vertices of can appears in different orders and thus Equation 3.1 follows.
We conclude from Equation 3.1 that the number of full paths which contain nodes corresponding to a fixed monochromatic complete subgraph of size is at most

(3.2)

The total number of full paths in is

And hence the number of monochromatic complete subgraphs of size in is at least

(3.3)

One can prove by induction (see Lemma A.1 of Appendix A) that

(3.4)

We conclude from Equations 3.3 and 3.4 that the number of monochromatic complete subgraphs of size in is at least

(3.5)

Where the last inequality follows from the pentagonal number theorem (see Lemma A.2 in Appendix A). And thus we are done.    

In [Con12] (Theorem ) the following is proven.

Theorem 3.3.

Let be natural numbers. Then, in any red/blue colouring of the edges of , there are at least

red monochromatic complete subgraphs of size or at least

blue monochromatic complete subgraphs of size .

For Theorem in [Con12] implies the following bound.

Theorem 3.4.

Let be a natural number. Then, in any red/blue colouring of the edges of , there are at least

monochromatic complete subgraphs of size .

We prove the following non-asymptotic version of Theorem 3.4.

Theorem 3.5.

Given a natural number and a natural number , Any -coloring of the edges of the complete graph on vertices contains at least

monochromatic complete subgraphs of size .

Proof.

The proof method is taken from the paper [Erd62]. Let be a graph on vertices and suppose . Let . By Theorem 3.2 every induced subgraph of vertices in contains at least monochromatic complete subgraphs of size . Notice that graph has induced subgraphs of size . Furthermore each monochromatic complete subgraph of size is contained in at most induced subgraphs of size . Hence graph contains at least

(3.6)

monochromatic complete subgraphs of size . Now as we conclude that graph contains at least

monochromatic complete subgraphs of size . And we are done.    

Theorem 3.6.

Any -coloring of the edges of the complete graph on vertices contains at least monochromatic complete subgraphs of size where

Proof.

Let be a -coloring of the edges of the complete graph on vertices. Let be the Ramsey Tree of as defined in Section 2.3. Henceforth we will denote a path of length (that is a path from a root node to a node in level ) in as a full path. Set

Notice that . Recall that any set of red nodes in a full path induces a red monochromatic complete subgraph in and any set of blue nodes in a full path induces a blue monochromatic complete subgraph in . If there are at least different monochromatic complete subgraphs of size induced by the paths of the Ramsey Tree we are done.

Thus we may assume that the number of different monochromatic complete subgraphs of size induced by the paths of the Ramsey Tree is at most . Given a fixed monochromatic complete subgraph of size , there are at most

(3.7)

different full paths of the Ramsey Tree which induce the monochromatic complete subgraph . This follows from the fact that we can choose the levels in which nodes corresponding to appears in in ways and in these levels can appear in at most different orders, now the product of the sizes of the bags in which the vertices of do not appear is at most

and bound 3.7 follows. By Lemma 2.7 the number of full paths in is . Hence if we will remove all the full paths which induce a monochromatic complete subgraph of size then we will remain with at least

(3.8)

full paths where each such full path induces two monochromatic complete subgraphs and such that and .
The vertices corresponding to a full path can appear in at most (t+1)! orders. Hence we have at least full paths where no two such paths correspond to the same set of vertices in . Thus no two such paths can induce the same two monochromatic complete subgraphs and . Hence we have at least

(3.9)

monochromatic complete subgraphs where each such subgraph is of size at least and at most . This concludes the proof.    

Now we shall show that the bounds in Theorem 3.6 can be slightly improved (with a more complicated proof).

Theorem 3.7.

For large enought , any -coloring of the edges of the complete graph on vertices contains at least monochromatic complete subgraphs of size where

Proof.

A warning to the reader: it is required in the proof that certain numbers are integers though we have omitted the corresponding floor/ceiling brackets. Since we do not expect any confusion this will hopefully improve the readability of the proof and of course it will not effect the asymptotics.
Let be a -coloring of the edges of the complete graph on vertices. Let be the Restricted Ramsey Tree (RRT) of as defined in Section 2.4, where each node in has bias . By the definition of the RRT we have that the bags of the nodes in level of the RRT (where ) are of size , that is (recall also that we defined ). Denote by the number of monochromatic complete subgraphs of size in .

  • Let be the set of the indices of the levels of .

  • Let be the set of the indices of the first levels of