  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.
