On the Off-Diagonal Unordered Erdős-Rado Numbers

Abstract

Erdős and Rado [Journal of the London Mathematical Society 25 (1950)] introduced the Canonical Ramsey numbers $\text{er}(t)$ as the minimum number $n$ such that every edge-coloring of the ordered complete graph $K_n$ contains either a monochromatic, rainbow, upper lexical, or lower lexical clique of order $t$. Richer [Journal of Combinatorial Theory Series B 80 (2000)] introduced the unordered asymmetric version of the Canonical Ramsey numbers $\text{CR}(s,r)$ as the minimum $n$ such that every edge-coloring of the (unordered) complete graph $K_n$ contains either a rainbow clique of order $r$, or an orderable clique of order $s$.

We show that $\text{CR}(s,r) = O(r^3/\log r)^{s-2}$, which, up to the multiplicative constant, matches the known lower bound and improves the previously best known bound $\text{CR}(s,r) = O(r^3/\log r)^{s-1}$ by Jiang [Discrete Mathematics 309 (2009)]. We also obtain bounds on the further variant $\text{ER}(m,\ell,r)$, defined as the minimum $n$ such that every edge-coloring of the (unordered) complete graph $K_n$ contains either a monochromatic $K_m$, lexical $K_\ell$, or rainbow $K_r$.