A combinatorial proof of the Burdzy–Pitman conjecture

Abstract

We prove a sharp upper bound for the number of high degree differences in bipartite graphs: let ( U, V, E ) be a bipartite graph with U = { u 1 , u 2 , . . . , u n } and V = { v 1 , v 2 , . . . , v n } ; for n ≥ k > n 2 we show that As a direct application we show a slightly stronger, probabilistic version of this theorem and thus confirm the Burdzy–Pitman conjecture about the maximal spread of coherent and independent distributions.