2026/07/19 by Anurag Bishnoi, Gaurav Kucheriya
#math.CO
For every fixed integer t ≥ 2, we give an upper bound on the multicolor vector space Ramsey number R2(t; k) that is a tower function of height independent of k. For t ≥ 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 R2(t; k) ≤ \lceil log R(Ks(r); k + 1) \rceil ≤ twrr-1(c klog k), for r = 2t - 1 and s = 2t - 1, where R(Ks(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.