Kohayakawa–Kreuter conjecture on asymmetric Ramsey thresholds

Let $H_1,\dots,H_r$ be graphs. For a graph $H$, write $m_2(H)$ for its 2-density, and, when $m_2(H_1)\geq m_2(H_2)$, define the mixed 2-density by

$$m_2(H_1,H_2)=\max\left\{\frac{{\mathsf{e}}(J)}{{\mathsf{v}}(J)-2+1/m_2(H_2)}:J\subseteq H_1,\ {\mathsf{v}}(J)\geq2\right\}.$$

Let $G_{n,p}$ be the binomial random graph, and say that it is Ramsey for $(H_1,\dots,H_r)$ if every $r$-edge-colouring contains a monochromatic copy of $H_i$ in colour $i$. Kohayakawa–Kreuter conjecture. If $m_2(H_1)\geq\dotsb\geq m_2(H_r)$ and $m_2(H_2)>1$, then there exist constants $C>c>0$ such that

$$\lim_{n\to\infty}\operatorname{Pr}(G_{n,p}\text{ is Ramsey for }(H_1,\dots,H_r))= \begin{cases} 1&\text{if }p\geq Cn^{-1/m_2(H_1,H_2)},\\ 0&\text{if }p\leq cn^{-1/m_2(H_1,H_2)}. \end{cases}$$

The paper confirms this conjecture for all $r$-tuples of graphs, so it is solved; the mixed 2-density identifies the threshold scale for asymmetric random Ramsey properties.