Upper bounds on Ramsey numbers for vector spaces over finite fields

Abstract

For $B \subseteq \mathbb F_q^m$, let $\exaff(n,B)$ denote the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B \subseteq \mathbb F_q^m$, $\exaff(n,B)=o(q^n)$ as $n \to \infty$. For certain $q$ and $B$, some more precise bounds are known. We connect some of these problems to certain Ramsey-type problems, and obtain some new bounds for the latter. For $s,t \geq 1$, let $R_q(s,t)$ denote the minimum $n$ such that in every red-blue coloring of one-dimensional subspaces of $\mathbb F_q^n$, there is either a red $s$-dimensional subspace of $\mathbb F_q^n$ or a blue $t$-dimensional subspace of $\mathbb F_q^n$. The existence of these numbers is implied by the celebrated theorem of Graham, Leeb, Rothschild. We improve the best known upper bounds on $R_2(2,t)$, $R_3(2,t)$, $R_2(t,t)$, and $R_3(t,t)$.