Georgia Institute of Technology College of Sciences

Turán-type problems ask for the densest-possible structure which avoids a fixed substructure H. Ramsey-type problems ask for the largest possible "complete" structure which can be decomposed into a fixed number of H-free parts. We discuss some of these problems in the context of vector spaces over finite fields. In the Turán setting, Furstenberg and Katznelson showed that any constant-density subset of the affine space AG(n,q) must contain a k-dimensional affine subspace if n is large enough. On the Ramsey side of things, a classical result of Graham, Leeb, and Rothschild implies that any red-blue coloring of the projective space PG(n-1,q) must contain a monochromatic k-dimensional projective subspace, for n large. We highlight the connection between these results and show how to obtain new bounds in the latter (projective Ramsey) problem from bounds in the former (affine Turán) problem. This is joint work with Liana Yepremyan.