The Erdős–Szekeres Theorem and Congruences

The following problem of combinatorial geometry is considered. Given positive integers $n$ and $q$, find or estimate a minimal number $h$ for which any set of $h$ points in general position in the plane contains $n$ vertices of a convex polygon for which the number of interior points is divisible by $q$. For a wide range of parameters, the existing bound for $h$ is dramatically improved.