Szemerédi’s Regularity Lemma and Quasi-randomness

The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of ‘pseudorandomness’ or ‘quasi-randomness’. We then state two closely related variants of the regularity lemma for sparse graphs and present a proof for one of them. In the second half of the paper, we discuss a basic idea underlying the algorithmic version of the original regularity lemma: we discuss a ‘local’ condition on graphs that turns out to be, roughly speaking, equivalent to the regularity condition of Szemerédi. Finally, we show how the sparse version of the regularity lemma may be used to prove the equivalence of a related, local condition for regularity. This new condition turns out to give a O(n2) time algorithm for testing the quasi-randomness of an n-vertex graph.