Multicolor vector space Ramsey numbers over the binary field

For every fixed integer $t\ge 2$, 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 $t\ge 3$, 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 $$R_2(t;k)\le \lceil \log R(K_s^{(r)};k+1)\rceil \le {\mathrm{t}\mathrm{w}\mathrm{r}}_{r-1}(ck\log k),$$ 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.