Cutting a graph into two dissimilar halves

Given a graph G and a subset S of the vertex set of G , the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S. We show that for every graph with n vertices and e edges, n < e < n ( n − 1)/4, there is an n /2‐element subset with the discrepancy of the order of magnitude of \documentclass{article}\pagestyle{empty}\begin{document}$\sqrt {ne}$\end{document} For graphs with fewer than n edges, we calculate the asymptotics for the maximum guaranteed discrepancy of an n /2‐element subset. We also introduce a new notion called “bipartite discrepancy” and discuss related results and open problems.