Threshold functions for patterns in random subsets of finite vector spaces

Changhao Chen, Catherine Greenhill

Published 2018-05-10Version 1

We study the existence of certain "patterns" in random subsets of vector spaces over finite fields. The patterns we consider are three-term arithmetic progressions, right triangles, parallelograms and affine planes. We give a threshold function for the property that a random subset of vectors contains a pattern from a given family, and show that the number of patterns contained in the random subset is asymptotically Poisson at the threshold scale.