Discrepancy in graphs and hypergraphs

Let G be a graph with n vertices and p ( n 2 ) edges, and define the discrepancies discp (G) = maxY⊂V (G) { e(Y )− p (|Y | 2 )} and discp (G) = maxY⊂V (G) { p (|Y | 2 ) − e(Y ) } . We prove that if p(1 − p) ≥ 1/n then discp (G)disc − p (G) ≥ p(1 − p)n3/6400. We also prove a similar inequality for k-uniform hypergraphs, and give related results concerning 2-colourings of k-uniform hypergraphs. Our results extend those of Erdős, Goldberg, Pach and Spencer [6] and Erdős and Spencer [7].