The Number of Edge Colorings With No Monochromatic Cliques

Abstract

Let F (n, r, k) denote the maximum possible number of distinct edge-colorings of a simple graph on n vertices with r colors which contain no monochromatic copy of K k . It is shown that for every fixed k and all n > n 0 (k), F (n, 2, k) = 2 t k −1 (n) and F (n, 3, k) = 3 t k −1 (n) , where t k−1 (n) is the maximum possible number of edges of a graph on n vertices with no K k (determined by Turán's theorem). The case r = 2 settles an old conjecture of Erdős and Rothschild, which was also independently raised later by Yuster. On the other hand, for every fixed r > 3 and k > 2, the function F (n, r, k) is exponentially bigger than r t k −1 (n) . The proofs are based on Szemerédi's regularity lemma together with some additional tools in extremal graph theory, and provide one of the rare examples of a precise result proved by applying this lemma.

The Number Of Orientations Having No Fixed Tournament

“…Our approach in the proof of Lemma 2.1 is similar to the one from [2] and [3], which is based on two important tools, the Simonovits stability theorem and the Szemerédi regularity lemma. However, we shall require a (somewhat uncommon) version of the regularity lemma for directed graphs and a few other additional ideas.…”

“…In section 4 we give a different proof for the special case T = C 3 that avoids using the regularity lemma, and obtain a moderate value for n 0 (C 3 ) (that can be optimized to less than 10000). Section 4 also contains a description of a simple reduction from the problem of counting the number or red-blue edge colorings of a graph G having no monochromatic K k (solved in [11] for k = 3 and in [2] for k > 3) to the problem of counting the number of orientations of a graph G that do not contain the transitive tournament on k vertices, denoted T k . Using this reduction we show, in particular, that n 0 (T 3 ) = 1.…”

“…It is based on the basic approach in [2] with some additional ideas, and uses several tools from Extremal Graph Theory, including a (somewhat uncommon) directed version of the regularity lemma of Szemerédi. It provides a rare example in which this lemma is used to prove results on directed graphs, and an even more rare example of a precise result obtained with the lemma.…”

On A Hypergraph Turán Problem Of Frankl

“…The process of separating the argument into a stability stage and a refinement stage focuses attention on the particular difficulties of each, and often leads to progress where the raw problem has appeared intractable. For recent examples we refer to our proofs of the conjecture of Sós on the Turán number of the Fano plane [4], and a conjecture of Yuster on edge colorings with no monochromatic cliques [1].…”

“…Looking at the vertices of such an edge in some order, we can select the first 2 vertices in αn(αn − 1) ways. Since the edge is good, the choice of 2 vertices together with a restricts the fourth vertex to lie in some particular class V i , so it can be chosen in at most 1 2 + 10 −6 n ways. Note that we have counted each edge 6 times, so we get at most αn(αn…”