Ramsey Goodness of Paths in Random Graphs

We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In this paper we study the threshold for the event that the Erd\H{o}s--R\'enyi random graph $G(N,p)$ is Ramsey for a clique versus a path. We show that $$G\big( (1 + \varepsilon) rn,p \big) \to (K_{r+1},P_n)$$ with high probability if $p \gg n^{-2 / (r + 1)}$, and $$G\big( rn + t, p \big) \to (K_{r+1},P_n)$$ with high probability if $p \gg n^{-2 / (r + 2)}$ and $t \gg 1/p$. Both of these results are sharp (in different ways), since with high probability $G(Cn,p) \not\to (K_{r+1}, P_n)$ for any constant $C > 0$ if $p \ll n^{-2/(r + 1)}$, and $G(rn + t, p) \not\to (K_{r+1}, P_n)$ if $t \ll 1/p$, for any $0 < p \le 1$.