2023/09/05 by Zach Hunter, Cosmin Pohoata, Hunter, Zach +1
Mathematics · #Limits and Structures in Graph Theory #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2309.02424
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 ≥ cd7 and any red-blue coloring of the one-dimensional subspaces of \mathbbF2n, 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 ⊂ \mathbbF2n with density α∈ [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.