On a generalization of the Hadwiger-Nelson problem

TL;DR

This paper generalizes the Hadwiger-Nelson problem by studying quadratic graphs over local fields, proving conditions for infinite Borel chromatic number, and applying spectral bounds and oscillatory integral analysis.

Contribution

It establishes a criterion for the Borel chromatic number of quadratic graphs over local fields, linking it to the quadratic form’s properties, and addresses a related combinatorial question.

Findings

Borel chromatic number is infinite iff the quadratic form represents zero non-trivially.

Spectral bounds for Cayley graphs are used to analyze chromatic properties.

Results extend the understanding of unit-distance graphs to quadratic forms over local fields.

Abstract

For a field $F$ and a quadratic form $Q$ defined on an $n$-dimensional vector space $V$ over $F$, let $\mathrm{QG}_Q$, called the quadratic graph associated to $Q$, be the graph with the vertex set $V$ where vertices $u,w \in V$ form an edge if and only if $Q(v-w)=1$. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In the present paper, we will prove that for a local field $F$ of characteristic zero, the Borel chromatic number of $\mathrm{QG}_Q$ is infinite if and only if $Q$ represents zero non-trivially over $F$. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009.