Multicolor vector space Ramsey numbers over the binary field

Abstract

For every fixed integer $t \geq 2$, we give an upper bound on the multicolor vector space Ramsey number $R_2(t; k)$ that is a tower function of height independent of $k$. For $t \geq 3$, this is the first bound of its form, significantly improving upon the earlier bounds that are towers of height linear in $k$. We achieve this by reducing the problem to a classical hypergraph Ramsey problem via binary simplex codes. In particular, we prove that $$R_2(t; k) \leq \left\lceil \log R(K_s^{(r)}; k + 1) \right\rceil \leq \mathrm{twr}_{r-1}(c k\log k),$$ for $r = 2^{t - 1}$ and $s = 2^t - 1$, where $R(K_{s}^{(r)}; k + 1)$ is the classical $(k + 1)$-color Ramsey number for the complete $r$-uniform hypergraph on $s$ vertices. This improvement also translates into an improved lower bound on the chromatic number of the binary projective space with respect to $(t - 1)$-flats. For $t = 2$, it recovers the connection with multicolor Ramsey numbers for triangles.