On problems in extremal multigraph theory

TL;DR

This paper investigates extremal properties of multigraphs with constraints on edges among vertex subsets, advancing understanding of sum and product maximization problems with implications for hypergraph theory.

Contribution

It resolves key conjectures and reveals complex behaviors in extremal multigraph problems, highlighting potential computational intractability.

Findings

Resolved conjectures of Day, Falgas-Ravry, and Treglown.

Established intricate behaviors for sum and product extremal problems.

Provided evidence of potential computational intractability.

Abstract

A multigraph G is said to be an (s,q)-graph if every s-set of vertices in G supports at most q edges (counting multiplicities). In this paper we consider the maximal sum and product of edge multiplicities in an (s,q)-graph on n vertices. These are multigraph analogues of a problem of Erd\H{o}s raised by F\"uredi and K\"undgen and Mubayi and Terry respectively, with applications to counting problems and extremal hypergraph theory. We make major progress, settling conjectures of Day, Falgas-Ravry and Treglown and of Falgas-Ravry, establishing intricate behaviour for both the sum and the product problems, and providing both a general picture and evidence that the problems may prove computationally intractable in general.