9.4 Szemerédi 's Regularity Lemma

In this section we describe a fundamental result, the Regularity Lemma, proved by Endre Szemerédi in the 70s. The original motivation for proving it has been an application in Combinatorial Number Theory, leading, together with several additional deep ideas, to a complete solution of the Erdős-Turán conjecture discussed in Appendix B.2: every set of integers of positive upper density contains arbitrarily long arithmetic progressions. It took some time to realize that the lemma is an extremely powerful tool in Extremal Graph Theory, Combinatorics and Theoretical Computer Science. Stated informally, the regularity lemma asserts that the vertices of every large graph can be decomposed into a finite number of parts, so that the edges between almost every pair of parts form a random-looking graph. The power of the lemma is in the fact it deals with an arbitrary graph, making no assumptions, and yet it supplies much useful information about its structure. A detailed survey of the lemma and some of its many variants and fascinating consequences can be found in Komlós and Simonovits (1996). Let G = (V,E) be a graph. For two disjoint nonempty subsets of vertices A,B ⊂ V , let e(A,B) denote the number of edges of G with one end in A and one in B, and let d(A,B) = e(A,B) |A||B| denote the density of the pair (A,B). For a real ε > 0, a pair (A,B) as above is called ε-regular if for every X ⊂ A and Y ⊂ B that satisfy |X| ≥ ε|A|, |Y | ≥ ε|B| the inequality |d(A,B) − d(X,Y )| ≤ ε holds. It is not difficult to see that for every fixed positive ε, p, a fixed pair of two sufficiently large disjoint subsets A and B of a random graph G = G(n, p) are very likely to be ε-regular of density roughly p. (This is stated in one of the exercises at the end of the chapter.) Conversely, an ε-regular pair A,B with a sufficiently small positive ε is random-looking in the sense that it shares many properties satisfied by random (bipartite) graphs. A partition V = V0 ∪ V1 ∪ · · · ∪ Vk of V into pairwise disjoint sets in which V0 is called the exceptional set is an equipartition if |V1| = |V2| = · · · = |Vk|. We view the exceptional set as |V0| distinct parts, each consisting of a single vertex. For two partitions P and P ′ as above, P ′ is a refinement of P , if every part in P is a union of some of the parts of P ′. By the last comment on the exceptional set this means, in particular, that if P ′ is obtained from P by shifting vertices from the other sets in the partition to the exceptional set, then P ′ is a refinement of P . An equipartition is called ε-regular if |V0| ≤ ε|V | and all pairs (Vi, Vj) with 1 ≤ i < j ≤ k, except at most εk of them, are ε-regular.