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.
On the flip side, with regards to the lower bound in Inequality , in 1979 Giraud @cite_1 proved that @math . More recently in 2012, Conlon @cite_5 proved that there must exist at least @math monochromatic complete subgraphs of size @math , in any @math -colouring of the edges of @math , where @math and @math is a constant independent of @math . This result is incomparable with our Theorem .
@cite_5 @cite_1
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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 The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. A general theorem on the complexity of a class of recognition problems is proved. As a particular case the following result is given: There is no algorithm which, for any 2-coloration of the infinite complete graph, can produce a monochromatic subgraph of k vertices within 2k2 steps (at each step the color of an arbitrary edge is questioned). Ramsey's theorem guarantees that if G is a graph, then any 2-coloring of the edges of a large enough complete graph yields a monochromatic copy of G. Interesting problems arise when one asks how many such G must occur. A survey of this and related problems is given, along with a number of new results.