In this paper we investigate the behaviour of subgraphs of sparse ε-regular bipartite graphs G = (V1 ∪ V2, E) with vanishing density d that are induced by small subsets of vertices. In particular, we show that, with overwhelming probability, a random set S ⊆ V1 of size s 1/d contains a subset S′ with |S′| ≥ (1 − ε′)|S| that induces together with V2 an ε′-regular bipartite graph of density (1 ± ...
