An extremal graph problem with a transcendental solution

TL;DR

This paper demonstrates that the maximum number of certain multigraphs with a vertex set of size n, constrained by a four-vertex edge limit, grows exponentially with a transcendental base, revealing a novel transcendental solution in extremal graph theory.

Contribution

It introduces the first explicit extremal graph problem whose solution involves a transcendental number, using innovative methods like Zykov symmetrization and hypergraph containers.

Findings

Number of such multigraphs is a^{n^2 + o(n^2)} with transcendental a

Solution connects extremal graph theory to transcendental number theory

Employs novel symmetrization and induction techniques

Abstract

We prove that the number of multigraphs with vertex set $\{1, \ldots, n\}$ such that every four vertices span at most nine edges is $a^{n^2 + o(n^2)}$ where $a$ is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy-Schwarz arguments. Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.