Previous work on Ramsey multiplicity focused primarily on the multiplicity of complete graphs of constant size. We study the question for larger complete graphs, specifically showing that in every 2-coloring of a complete graph on @math vertices, there are at least @math monochromatic complete subgraphs of size between @math and @math . We also study bounds on the ratio between a maximum and a random red clique in a graph, and the ratio between a maximum and a random monochromatic clique in a graph.
 Contre-exemples a la conjecture de Erdos et discussion des proprietes des graphes extremaux Abstract Denote by k t ( G ) the number of cliques of order t in the graph G . Let k t (n)= min k t (G) + k t ( G ): |G| = n , where Ḡ denotes the complement of G , and | G | denotes the order of G . Let c t (n) = k t (n) ( n t , and let c t = lim n →∞ c t ( n ). An old conjecture of Erdos (1962), related to Ramsey's theorem, states that c t = 2 1−( t 2 ) . It was shown false by Thomason (1989) for all t ⩾4. We present a class of simply describable Cayley graphs which also show the falsity of Erdos conjecture for t = 4. These graphs were found by a computer search and, although of large orders (2 10 -2 14 ), they are rather simple and highly regular. The smallest upper bound for c 4 obtained by us is 0.976501 x 1 32 , and is given by the graph on the power set of a 10-element set (and, hence, of order 2 10 ) determined by the configuration 1,3,4,7,8,10 . and by the graph on the power set of 11 elements (and, hence, of order 2 11 ) determined by the configuration 1,3,4,7,8,10,11 . It is also shown that the ratio of edges to nonedges in a sequence contradicting the conjecture for t = 4 may approach 1, unlike in the sequences of graphs Thomason used in 1989.