On off-diagonal Ramsey numbers for vector spaces over ${\mathbb{F}}_2$

Abstract

For every positive integer $d$, we show that there must exist an absolute constant $c>0$ such that the following holds: for any integer $n\ge c{d}^7$ and any red-blue coloring of the one-dimensional subspaces of ${\mathbb{F}}_2^n$, there must exist either a $d$-dimensional subspace for which all of its one-dimensional subspaces get colored red or a $2$-dimensional subspace for which all of its one-dimensional subspaces get colored blue. This answers recent questions of Nelson and Nomoto, and confirms that for any even plane binary matroid $N$, the class of $N$-free, claw-free binary matroids is polynomially $\ddot{I}‡$-bounded. Our argument will proceed via a reduction to a well-studied additive combinatorics problem, originally posed by Green: given a set $A\subset {\mathbb{F}}_2^n$ with density $\hat{I}\pm \in [0,1]$, what is the largest subspace that we can find in $A+A$? Our main contribution to the story is a new result for this problem in the regime where $1/\hat{I}\pm$ is large with respect to $n$, which utilizes ideas from the recent breakthrough paper of Kelley and Meka on sets of integers without three-term arithmetic progressions.

incorporated referee comments; to appear in Math. Proc. Camb. Phil. Soc