A note on multidimensional Ramsey numbers

Fix integers $d,r\ge 2$ and suppose that the edge set of the $d$-fold Cartesian product of the $N$-clique $K_N^d$ is $r$-colored. We show that there is a copy of $K_n^d$ whose edges in each direction are monochromatic provided $N > 2^{2^{c n^{d-1}}}$, where $c$ depends only on $r$ and $d$. This improves the previous best exponent of $n^d$ proved by Gir\~ao, Kronenberg, and Scott while also improving the best known bound due to them for a multidimensional Erd\H os-Szekeres monotone subsequence theorem introduced by Fishburn and Graham.