A note on hypergraphs with asymmetric Ramsey properties

Let $r,\mathrm{\ell }\ge 2$ be integers. Given $r$-graphs $G$ and $F_1,\dots ,F_{\mathrm{\ell }}$, we write $G\to (F_1,\dots ,F_{\mathrm{\ell }})$ if every $\mathrm{\ell }$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$th color for some $1\le i\le \mathrm{\ell }$, otherwise we write $G\to ̸(F_1,\dots ,F_{\mathrm{\ell }})$. The Ramsey number $R(F_1,\dots ,F_{\mathrm{\ell }})$ is the minimum number of vertices in an $r$-graph $G$ satisfying $G\to (F_1,\dots ,F_{\mathrm{\ell }})$. In this note we prove that for any integers $t_1\ge \cdots \ge t_{\mathrm{\ell }}>r$, there exists an $r$-graph $G$ such that $G\to ̸(K_{t_1}^{(r)},\dots ,K_{t_{\mathrm{\ell }}}^{(r)})$ but $G\to (K_s^{(r)},K_{t_{\mathrm{\ell }}-1}^{(r)})$, where $s=R(K_{t_1}^{(r)},\dots ,K_{t_{\mathrm{\ell }}}^{(r)})-1$. This extends recent work by Mendonça, Miralaei, and Mota, who established the statement for $r=2$.