Chromatic thresholds for linear equations and recurrence

TL;DR

This paper investigates the chromatic thresholds for solution-free sets of linear equations over finite fields, establishing conditions for when these thresholds are zero and introducing new combinatorial and topological methods.

Contribution

It introduces the concept of chromatic thresholds for linear equations over finite fields, characterizes when these thresholds are zero, and develops new bounds using Kneser-type graphs and topological arguments.

Findings

Chromatic threshold is zero iff the equation contains a zero-sum subcollection of at least three coefficients.

Established a quantitative chromatic lower bound for Cayley graphs generated by Hamming balls.

Resolved a question of Griesmer and related recurrence properties in abelian groups.

Abstract

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study $\mathcal L$-solution-free sets $A\subseteq \mathbb F_p$ through the chromatic number of the Cayley graph $\mathsf{Cay}(\mathbb F_p,A)$. We introduce the \emph{chromatic threshold} $\delta_\chi(\mathcal L)$, the minimum density that guarantees bounded chromatic number of $\mathsf{Cay}(\mathbb F_p,A)$ among all $\mathcal L$-solution-free sets $A$, and determine exactly when $\delta_\chi(\mathcal L)=0$. We prove that $\delta_\chi(\mathcal L)=0$ if and only if $\mathcal L$ contains a zero-sum subcollection of at least three coefficients. A key ingredient is a quantitative chromatic lower bound for Cayley graphs on $\mathbb Z_p^n$ generated by Hamming balls around the all-ones vector. This is obtained by introducing a new Kneser-type graph that admits a natural embedding into $\mathbb Z_p^n$, together with an equivariant Borsuk--Ulam type argument. As a consequence, we resolve a question of Griesmer. We further relate our classification to the hierarchy of measurable, topological, and Bohr recurrence. In particular, we show that every infinite discrete abelian group admits a set that is topological recurrent but not measurable recurrent, extending the seminal examples of K\v{r}\'i\v{z} and Ruzsa.