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 $χ$-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 $α\in [0,1]$, 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 $1/α$ 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.