Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II

Chi Hoi Yip

TL;DR

This work gives a new, streamlined proof of Blokhuis' theorem and its extensions to maximum cliques in Cayley graphs over finite fields, framed through additive combinatorics and finite geometry rather than heavy analytic tools. By analyzing the multiplicative doubling of the connection set $SalanceF_{q^2}^*$ and employing a subfield criterion tied to directions in affine planes, the authors show that any clique of size $q$ with $0,1alance A$ and $A-Aalance Sigcupig\{0igig)$ must be the subfield $F_q$, or else all maximum cliques have a subfield-affine structure $aF_q+b$ with $aalance S$. This yields a dichotomy for Cayley graphs and recovers, unifies, and strengthens results for generalized Paley graphs $GP(q^2,d)$, including new regimes where $d mid (q+1)$ and when $S$ is a union of cosets of a fixed subgroup. Overall, the paper broadens the scope of vLM-type results using additive-combinatorial methods and affine-geometry tools, eliminating reliance on deep number-theoretic sums and providing explicit structural conclusions on maximum cliques.

Abstract

The well-known Van Lint--MacWilliams' conjecture states that if $q$ is an odd prime power, and $A\subseteq \mathbb{F}_{q^2}$ such that $0,1 \in A$, $|A|=q$, and $a-b$ is a square for each $a,b \in A$, then $A$ must be the subfield $\mathbb{F}_q$. This conjecture was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if $S \subseteq \mathbb{F}_{q^2}^*$ has small multiplicative doubling, and $A\subseteq \mathbb{F}_{q^2}$ with $0,1 \in A$, $|A|=q$, such that $A-A \subseteq S \cup \{0\}$, then $A=\mathbb{F}_q$. This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.

Van Lint-MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II

TL;DR

This work gives a new, streamlined proof of Blokhuis' theorem and its extensions to maximum cliques in Cayley graphs over finite fields, framed through additive combinatorics and finite geometry rather than heavy analytic tools. By analyzing the multiplicative doubling of the connection set and employing a subfield criterion tied to directions in affine planes, the authors show that any clique of size with and must be the subfield , or else all maximum cliques have a subfield-affine structure with . This yields a dichotomy for Cayley graphs and recovers, unifies, and strengthens results for generalized Paley graphs , including new regimes where and when is a union of cosets of a fixed subgroup. Overall, the paper broadens the scope of vLM-type results using additive-combinatorial methods and affine-geometry tools, eliminating reliance on deep number-theoretic sums and providing explicit structural conclusions on maximum cliques.

Abstract

The well-known Van Lint--MacWilliams' conjecture states that if is an odd prime power, and such that , , and is a square for each , then must be the subfield . This conjecture was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if has small multiplicative doubling, and with , , such that , then . This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.
Paper Structure (6 sections, 8 theorems, 18 equations)

This paper contains 6 sections, 8 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Directions determined by a point set in an affine plane
  3. Proof of main results
  4. A criterion for subfields
  5. Proof of Theorem \ref{['thm:main']}
  6. Proof of Theorem \ref{['thm:GP']}

Key Result

Theorem 1.1

Let $S \subseteq \mathbb{F}_{q^2}^*$ with $S=-S$ such that If $A\subseteq \mathbb{F}_{q^2}$ with $|A|=q$ and $0,1\in A$ such that $A-A \subseteq S \cup \{0\}$, then $A$ is the subfield $\mathbb{F}_q$. Equivalently, if $A\subseteq \mathbb{F}_{q^2}$ is a clique in $X=\operatorname{Cay}(\mathbb{F}_{q^2}; S)$ with $|A|=q$ and $0,1\in A$, then $A$ is the subfiel

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3: Blokhuis, Sziklai
  • Theorem 1.4
  • Theorem 2.1: Blokhuis, Ball, Brouwer, Storme, and Szőnyi
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 5 more