Ben Lund, Thresholds for incidence properties in finite vector spaces

Suppose that $E$ is a subset of ${\mathbb{F}}_q^n$, so that each point is contained in $E$ with probability $\theta$, independently of all other points. Then, what is the probability that there is an $m$-dimensional affine subspace that contains at least $\ell$ points of $E$? What is the probability that $E$ intersects all $m$-dimensional affine subspaces? We give Erdős-Renyi threshold functions for these properties, in some cases sharp thresholds. Our results improve previous work of Chen and Greenhill. This is joint work with Jeong Han Kim, Thang Pham, and Semin Yoo.