On off-diagonal Ramsey numbers for vector spaces over $\mathbb{F}_{2}$

TL;DR

This paper establishes bounds on off-diagonal Ramsey numbers for vector spaces over , linking combinatorics and additive number theory, and confirms polynomial hi-boundedness for certain binary matroids.

Contribution

It provides a new bound for Ramsey numbers in vector spaces over  and reduces the problem to an additive combinatorics question, extending previous work.

Findings

Proves existence of monochromatic subspaces under certain conditions

Reduces a combinatorial problem to an additive combinatorics problem

Introduces a new result for sumset structure in ^n

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 $n \geq cd^{7}$ 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 $\chi$-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 $\alpha \in [0,1]$, 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 $1/\alpha$ 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.