Shengtong Zhang (张盛桐), Triangle Ramsey numbers of complete graphs

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that $$r_{K_3}(K_t)=(\genfrac{}{}{0ex}{}{r(K_t)}{3})$$ for all sufficiently large $t$.  Our proof involves a combination of results on the chromatic number of triangle-sparse graphs.

Joint work with Jacob Fox and Jonathan Tidor.