Extremal theory of locally sparse multigraphs

TL;DR

This paper determines the maximum product of edge multiplicities in certain multigraphs with local sparsity constraints, extending classical extremal graph theory results and exploring the complexity of the problem.

Contribution

It extends extremal graph theory to multigraphs by analyzing the maximum product of edge multiplicities under local sparsity constraints and establishes related stability and enumeration results.

Findings

Maximum product of edge multiplicities determined for specific congruence classes

Product-stability theorems established for $(n,s,q)$-graphs

Results extend classical extremal graph theory to multigraphs

Abstract

An $(n,s,q)$-graph is an $n$-vertex multigraph where every set of $s$ vertices spans at most $q$ edges. In this paper, we determine the maximum product of the edge multiplicities in $(n,s,q)$-graphs if the congruence class of $q$ modulo ${s\choose 2}$ is in a certain interval of length about $3s/2$. The smallest case that falls outside this range is $(s,q)=(4,15)$, and here the answer is $a^{n^2+o(n^2)}$ where $a$ is transcendental assuming Schanuel's conjecture. This could indicate the difficulty of solving the problem in full generality. Many of our results can be seen as extending work by Bondy-Tuza and F\"uredi-K\"undgen about sums of edge multiplicities to the product setting. We also prove a variety of other extremal results for $(n,s,q)$-graphs, including product-stability theorems. These results are of additional interest because they can be used to enumerate and to prove logical 0-1 laws for $(n,s,q)$-graphs. Our work therefore extends many classical enumerative results in extremal graph theory beginning with the Erd\H{o}s-Kleitman-Rothschild theorem to multigraphs.