Brendan Murphy

Solymosi's conjecture for rich lines in general position

We give an explicit construction that disproves a conjecture of Solymosi on the number of lines in general position that contain a near maximal number of points of a Cartesian product set. We will explain how this problem is connected to the sum-product problem, and show why Solymosi's conjecture is plausible. The counter-example we give is motivated by related questions in group theory, and is rather different from the usual examples in arithmetic combinatorics.

Brendan Murphy is a Heilbronn Fellow at the University of Bristol. He studies arithmetic combinatorics, specifically the sum-product problem and questions related to growth in groups.