@MISC{_1.preliminaries, author = {}, title = {1. Preliminaries}, year = {} }

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Abstract

V. Szemerédi’s regularity lemma The ‘regularity lemma ’ of Endre Szemerédi [5] roughly asserts that, for each ε> 0, there exists a number k such that the vertex set V of any graph G = (V, E) can be partitioned into at most k almost equal-sized classes so that between almost any two classes, the edges are distributed almost homogeneously. Here almost depends on ε. We will make this precise and prove it in Section 2. First, some ‘ε-free ’ preliminaries. Let G = (V, E) be a graph. For nonempty A, B ⊆ V, define (1) e(A, B): = number of adjacent pairs in A × B, d(A, B):= e(A, B)