Distance graphs in vector spaces over finite fields, coloring and pseudo-randomness

In this paper we systematically study various properties of the distance graph in ${\mathbb{F}}_q^d$, the $d$-dimensional vector space over the finite field ${\mathbb{F}}_q$ with $q$ elements. In the process we compute the diameter of distance graphs and show that sufficiently large subsets of $d$-dimensional vector spaces over finite fields contain every possible finite configurations.