The general off-diagonal unordered Erdős–Rado bound

For integers $m,\ell\geq 4$, let $\operatorname{ER}(m,\ell,r)$ denote the off-diagonal unordered Erdős–Rado number for monochromatic and lexical cliques of the indicated orders against a rainbow clique of order $r$. The general off-diagonal Erdős–Rado conjecture. For every $m,\ell\geq 4$, there exists a constant $c=c(m,\ell)>0$ such that, for every $r\geq 3$,

$$\operatorname{ER}(m,\ell,r)\leq c\left(\frac{r^3}{\log r}\right)^{\ell-2}.$$

The paper establishes the correct asymptotics in several boundary cases, including $\operatorname{ER}(m,3,r)$ and $\operatorname{ER}(3,\ell,r)$, but leaves a gap for general $m,\ell\geq4$; the conjecture asserts that the known lower-bound order is sharp.