Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs

Abstract

Letk t(G) be the number of cliques of ordert in the graphG. For a graphG withn vertices let $$c_t (G) = \frac{{k_t (G) + k_t (\bar G)}}{{\left( {\begin{array}{*{20}c} n \\ t \\ \end{array} } \right)}}$$ . Letc t(n)=Min{c t(G)??G?=n} and let $$c_t = \mathop {\lim }\limits_{n \to \infty } c_t (n)$$ . An old conjecture of Erdos [2], related to Ramsey's theorem states thatc t=21-(t/2). Recently it was shown to be false by A. Thomason [12]. It is known thatc t(G)?21-(t/2) wheneverG is a pseudorandom graph. Pseudorandom graphs -- the graphs "which behave like random graphs" -- were inroduced and studied in [1] and [13]. The aim of this paper is to show that fort=4,c t(G)?21-(t/2) ifG is a graph arising from pseudorandom by a small perturbation.