Monochromatic triangles with empty intersection and Kneser Ramsey numbers

Recently, Heath, McCourt, Parker, Schwieder, and Zerbib initiated the systematic study of the $r$-Kneser Ramsey number $R_r^{KG}(s,t)$ and investigated related Ramsey-type problems. A central motivation for their work comes from a question of Holmsen, Hrusak, and Roldán-Pensado, who asked whether, for $n=2k-1$ and sufficiently large $k$, every red/blue edge-coloring of the complete graph on the vertex set $V=\left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ necessarily contains a monochromatic triangle $ABC$ with $A,B,C\in V$ and $A\cap B\cap C=\mathrm{\varnothing }$. Heath, McCourt, Parker, Schwieder, and Zerbib established that this conclusion holds when $n\ge \frac{7k}{3}$ and $k\ge 12$. We make substantial progress toward the problem of Holmsen, Hrusak, and Roldán-Pensado by proving that the conclusion already holds for every $k\ge 2$ whenever $n\ge 2k+1$. In addition, we obtain improved lower bounds for $R_r^{KG}(s,t)$ when $s$ and $t$ are fixed and $r$ is sufficiently large.