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 ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ 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. 
Derrick Hart, Alex Iosevich, Doowon Koh, Steve Senger and Ignacio   Uriarte-Tuero