On a generalization of the Hadwiger–Nelson problem

Abstract

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QG Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ 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 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 [6].