Szemerédi's partition and quasirandomness

In this paper we shall investigate the connection between the SzemerCdi Regularity Lemma and quasirandom graph sequences, defined by Chung, Graham, and Wilson, and also, slightly differently, by Thomason. We prove that a graph sequence (G,) is quasirandom if and only if in the SzemerCdi partitions of G, almost all densities are $ + o(1). Many attempts have been made to clarify when an individual event could be called random and in what sense. Both the fundamental problems of probability theory and some practical application need this clarification very much. For example, in applications of the Monte-Carlo method one needs to know if the random number generator used yields a sequence which can be regarded “pseudorandom” or not. The literature on this question is extremely extensive. Thomason [6-81 and Chung, Graham, and Wilson [2,3], and also Frankl, Rodl, and Wilson [4] started a new line of investigation, where (instead of regarding numerical sequences) they gave some characterizations of “randomlike” graph sequences, matrix sequences, and hypergraph sequences. The aim of this paper is to contribute to this question in case of graphs, continuing the above line of investigation. Let %(n, p) denote the probability space of labelled graphs on n vertices, where the edges are chosen independently and at random, with probability p. We shall say that “a random graph sequence (G,) has property P”