A characterization for sparse ε-regular pairs

We are interested in (ε)-regular bipartite graphs which are the central objects in the regularity lemma of Szemerédi for sparse graphs. A bipartite graph G = (A]B,E) with density p = |E|/(|A||B|) is (ε)-regular if for all sets A′ ⊆ A and B′ ⊆ B of size |A′| ≥ ε|A| and |B ′| ≥ ε|B|, it holds that |eG(A, B′)/(|A′||B′|) − p| ≤ εp. In this paper we prove a characterization for (ε)-regularity. That is, we give a set of properties that hold for each (ε)-regular graph, and conversely if the properties of this set hold for a bipartite graph, then the graph is f(ε)-regular for some appropriate function f with f(ε) → 0 as ε → 0. The properties of this set concern degrees of vertices and common degrees of vertices with sets of size Θ(1/p) where p is the density of the graph in question.