On the off-diagonal unordered Erdős-Rado numbers

Abstract: Erd\H{o}s and Rado [P. Erd\H{o}s, R. Rado, A combinatorial theorem, Journal of the London Mathematical Society 25 (4) (1950) 249-255] 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 [D. Richer, Unordered canonical Ramsey numbers, Journal of Combinatorial Theory Series B 80 (2000) 172-177] 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 (unorderd) 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 [T. Jiang, Canonical Ramsey numbers and proporly colored cycles, Discrete Mathematics 309 (2009) 4247-4252]. 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 (unorderd) complete graph $K_n$ contains either a monochromatic $K_m$, lexical $K_\ell$, or rainbow $K_r$.