Vector space Ramsey numbers and weakly Sidorenko affine configurations

Abstract

For $B\subseteq {\mathbb{F}}_q^m$, the $n$-th affine extremal number of $B$ is 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 $B\subseteq {\mathbb{F}}_q^m$, the $n$-th affine extremal number of $B$ is $o(q^n)$ as $n\to \infty$. By counting affine homomorphisms between subsets of ${\mathbb{F}}_q^n$, we derive new bounds and give new proofs of some previously known bounds for certain affine extremal numbers. At the same time, we establish corresponding supersaturation results. We connect these bounds to certain Ramsey-type numbers in vector spaces over finite fields. For $s,t\ge 1$, let $R_q(s,t)$ denote the minimum $n$ such that in every red-blue coloring of the one-dimensional subspaces of ${\mathbb{F}}_q^n$, there is either a red $s$-dimensional subspace or a blue $t$-dimensional subspace of ${\mathbb{F}}_q^n$. The existence of these numbers is a special case of a well-known 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 $R_3(t,t)$.

19 pages, 1 figure

References in corpus (1)

• Green's sumset problem at density one half