A combinatorial proof of the Burdzy-Pitman conjecture

TL;DR

This paper provides a combinatorial proof establishing a sharp upper bound on the number of high degree differences in bipartite graphs, confirming the Burdzy-Pitman conjecture related to the spread of certain distributions.

Contribution

It introduces a novel combinatorial proof for the Burdzy-Pitman conjecture, linking graph degree differences to probabilistic distribution spreads.

Findings

Proves a sharp upper bound for degree differences in bipartite graphs.

Confirms the Burdzy-Pitman conjecture about distribution spread.

Provides a probabilistic version of the combinatorial bound.

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 $U=\{u_1, u_2, \dots, u_n\}$ and $V=\{v_1, v_2, \dots, v_n\}$; for $n\ge k>\frac{n}{2}$ we show that $\sum_{1\le i,j \le n} 1 {\Big\{|\text{deg}(u_i)-\text{deg}(v_j)|\ge k}\Big\} \le 2k(n-k).$ 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.