# Keivan Mallahi Karai

## "On the Borel chromatic number of a generalization of unit-distance graphs"

 Date: Wed, November 4, 2015 Time: 11:15 Place: Research I Seminar Room

Abstract: For a quadratic form $$Q$$ defined on an $$n$$-dimensional vector space V over $$\mathbb{R}$$, let $$\Gamma(Q)$$ be the graph with the vertex set $$V$$ where vertices $$u, w \in V$$ form an edge if and only if $$Q(v-w) = 1$$. These graphs may be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger-Nelson problem. In this talk, I will discuss my work (joint with M. Bardestani) showing that the Borel chromatic number of $$\Gamma(Q)$$ is infinite if and only if $$Q$$ is indefinite. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals using van der Corput lemma. Some applications and generalizations (over other fields) will also be briefly discussed.