Extremal problems for multigraphs

TL;DR

This paper investigates extremal multigraph problems focusing on maximizing the product of edge multiplicities under vertex set constraints, providing new bounds, settling specific conjectures, and developing useful analytical tools.

Contribution

It introduces a general lower bound construction for the maximum product problem, proves several cases including a conjecture of Mubayi and Terry, and analyzes asymptotic behavior for sparse multigraphs.

Findings

Established a conjectured asymptotic lower bound for many (s,q) pairs.

Settled the Mubayi and Terry conjecture for (s,q)=(4,6a+3) with a≥2.

Determined the asymptotic behavior for sparse multigraphs with q≤2binom{s}{2}.

Abstract

An $(n,s,q)$-graph is an $n$-vertex multigraph in which every $s$-set of vertices spans at most $q$ edges. Tur\'an-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry [An extremal problem with a transcendental solution, Combinatorics Probability and Computing 2019] posed the problem of determining the maximum of the product of the edge multiplicities in $(n,s,q)$-graphs. We give a general lower bound construction for this problem for many pairs $(s,q)$, which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the $(s,q)=(4,6a+3)$ case of the problem (for $a\geq2$); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for `sparse' multigraphs (i.e. when $q\leq 2\binom{s}{2}$). Finally we introduce some tools that are likely to be useful for attacking the problem in general.