Uniformity of extremal graph-codes

TL;DR

This paper explores the pseudorandom behavior of extremal graph-codes and related structures, highlighting their significance in combinatorics and the density polynomial Hales–Jewett conjecture.

Contribution

It demonstrates instances of pseudorandomness in extremal graph-codes and connects these phenomena to broader combinatorial problems.

Findings

Extremal graph-codes exhibit strong pseudorandom behavior.

Connections are established between extremal structures and the density polynomial Hales–Jewett conjecture.

The work advances understanding of structure and randomness in extremal combinatorial objects.

Abstract

It is an important fact that extremal discrete structures -- that is, discrete structures of maximal size among those that avoid certain configurations -- exhibit strong pseudorandom behavior. We present instances of this phenomenon in the context of \emph{graph-codes}, a notion put forth recently by Alon, as well as on problems related to the density polynomial Hales--Jewett conjecture.