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Chromatic thresholds for linear equations and recurrence

Hong Liu, Zhuo Wu, Ningyuan Yang, Shengtong Zhang

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} $δ_χ(\mathcal L)$, 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 $δ_χ(\mathcal L)=0$. We prove that $δ_χ(\mathcal L)=0$ 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 Kříž and Ruzsa.

Chromatic thresholds for linear equations and recurrence

Abstract

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over . Given a homogeneous equation with , we study -solution-free sets through the chromatic number of the Cayley graph . We introduce the \emph{chromatic threshold} , the minimum density that guarantees bounded chromatic number of among all -solution-free sets , and determine exactly when . We prove that if and only if contains a zero-sum subcollection of at least three coefficients. A key ingredient is a quantitative chromatic lower bound for Cayley graphs on 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 , 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 Kříž and Ruzsa.
Paper Structure (20 sections, 21 theorems, 178 equations, 1 figure)

This paper contains 20 sections, 21 theorems, 178 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Roth meets Erdős and Simonovits
  3. Applications in topological dynamics
  4. Related results and our approach
  5. Related results.
  6. Our approach.
  7. Preliminaries
  8. Topology
  9. Additive combinatorics
  10. Probabilistic methods
  11. Proof of \ref{['thm:main0']} $(b) \Rightarrow (a)$
  12. Generalized Kneser graphs
  13. A generalized Kneser graph and its chromatic number
  14. Embedding into Cayley graphs: Proof of \ref{['thm:cayley']}
  15. Proof of Theorem chi-lb
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\mathcal{L}:\sum_{i\in[k]}c_i x_i=0$ be a homogeneous linear equation with $k\ge 3$ and $c_1,\dots,c_k\in\mathbb Z\setminus\{0\}$, and let $p$ be prime. Then the following are equivalent:

Figures (1)

  • Figure 1: Roth degenerate $\subseteq$ Vanishing chromatic threshold $\subseteq$ Ramsey-Turán degenerate.

Theorems & Definitions (41)

  • Theorem 1.1: roth1954certain
  • Theorem 1.2: bucić2025ramseyturanvariantrothstheorem
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • proof
  • Theorem 2.2: Dold's Theorem for $\mathbb Z_p$
  • Lemma 2.3: Lebesgue number lemma
  • ...and 31 more