Nearly tight bounds for MaxCut in hypergraphs

TL;DR

This paper establishes nearly tight bounds for the maximum size of cuts in hypergraphs, improving previous results and confirming conjectures about the size of such cuts relative to random cuts.

Contribution

It proves an approximate version of a conjecture on hypergraph cuts and confirms a conjecture on linear hypergraphs, using a novel approach.

Findings

Existence of r-cuts exceeding random cuts by 5(m^{2/3-5}) for large k

Every linear hypergraph has an r-cut exceeding random by 5(m^{3/4}), which is tight

Improved bounds confirm conjectures and extend previous results in hypergraph cut theory

Abstract

An $r$-cut of a $k$-uniform hypergraph is a partition of its vertex set into $r$ parts, and the size of the cut is the number of edges which have at least one vertex in each part. The study of the possible size of the largest $r$-cut in a $k$-uniform hypergraph was initiated by Erd\H{o}s and Kleitman in 1968. For graphs, a celebrated result of Edwards states that every $m$-edge graph has a $2$-cut of size $m/2+\Omega(m^{1/2})$, which is sharp. In other words, there exists a cut which exceeds the expected size of a random cut by the order of $m^{1/2}$. Conlon, Fox, Kwan and Sudakov proved that any $k$-uniform hypergraph with $m$ edges has an $r$-cut whose size is $\Omega(m^{5/9})$ larger than the expected size of a random $r$-cut, provided that $k \geq 4$ or $r \geq 3$. They further conjectured that this can be improved to $\Omega(m^{2/3})$, which would be sharp. Recently, R\"aty and Tomon improved the bound $m^{5/9}$ to $m^{3/5-o(1)}$ when $r \in \{ k-1,k\}$. Using a novel approach, we prove the following approximate version of the Conlon-Fox-Kwan-Sudakov conjecture: for each $\varepsilon>0$, there is some $k_0=k_0(\varepsilon)$ such that for all $k>k_0$ and $2\leq r\leq k$, in every $k$-uniform hypergraph with $m$ edges there exists an $r$-cut exceeding the random one by $\Omega(m^{2/3-\varepsilon})$. Moreover, we show that (if $k\geq 4$ or $r\geq 3$) every $k$-uniform linear hypergraph has an $r$-cut exceeding the random one by $\Omega(m^{3/4})$, which is tight and proves a conjecture of R\"aty and Tomon.