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.
We show that, for n large, there must exist at least @math monochromatic K t s in any two-colouring of the edges of K n , where C≈2.18 is an explicitly defined constant. The old lower bound, due to Erdős [2], and based upon the standard bounds for Ramsey’s theorem, is @math Resume Dans cet article qui developpe une Note ( Giraud, G., C. R. Acad. Sci. Paris Ser. A , 276 (1973) 1173–1175), nous prouvons que, pour n assez grand, tout bicoloriage des aretes de K n contient une proportion de K 4 dont les six aretes sont de meme couleur au moins egale a 1 46 , et que si le nombre de triangles unicolores est minimal, cette proportion passe au moins a (5 − 10 1 2 ) 64 a O(n 7 2 ) pres. Nous exposons aussi une methode de majoration des nombres de Ramsey binaires-bicolores qui fournit ϱ(6, 6) ≤ 169.