Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials
We introduce a robust OR polynomial framework for composing positive semidefinite certificates across OR constraints. We demonstrate the power of this method in two applications. The first is on acute-free families. A set $\mathcal F=\{(x_i^{(1)},\ldots,x_i^{(r)})\}_{i=1}^M \subseteq (S^{n-1})^r$ is $r$-way acute-free if, for every $i\neq j$, there is a coordinate $t\in[r]$ such that $\langle x_i^{(t)},x_j^{(t)}\rangle\leq 0$. We write $M_r(n)$ for the maximum size of such a set, and $M_r^{\pm}(n)$ for the hypercube restriction. On the hypercube, $r$-way acute-free sets are independent sets for some strong power graph $G_n^{\boxtimes r}$. The Lov\'asz theta number $\vartheta(G_n)$ is multiplicative but exponentially loose, whereas the Schrijver number $\vartheta'(G_n)$ gives the correct order, but is not multiplicative. We bypass this obstruction by proving a general quasi-tensorization result for the Schrijver number. That is, for every collection of graphs $G_1,\ldots, G_r$ satisfying $\vartheta'(G_i)\geq 2$, there is an absolute constant $C$ such that $\vartheta'(G_1\boxtimes \cdots \boxtimes G_r) \leq \prod_{i=1}^r \vartheta'(G_i)^{C\log r \log \vartheta'(G_i)}$. Applying this result gives that $M_r^{\pm}(n) \le M_r(n) \le (2n)^{C_0 r\log r\log(2n)}$ for some absolute constant $C_0$. The second application is on multicolor Ramsey numbers. The $r$-color Ramsey number $R_r(k)$ is the minimum $n$ such that every $r$-coloring of the edges of the complete graph on $n$ vertices contains a monochromatic copy of $K_k$. In a breakthrough result, Balister et al. [arXiv:2410.17197] showed that $R_r(k)\le \exp(-\Omega(k/r^{12}))r^{rk}$ via a geometric lemma. By improving the $r$ dependency in their geometric lemma via the OR polynomial framework, we prove that $R_r(k)\le \exp(-\Omega(k/(r^9(\log r)^6)))r^{rk}$.