The anti-Ramsey threshold of complete graphs

Abstract

For graphs $G$ and $H$, let $G {\displaystyle\smash{\begin{subarray}{c} \hbox{$\tiny\rm rb$} \\ \longrightarrow \\ \hbox{$\tiny\rm p$} \end{subarray}}}H$ denote the property that for every proper edge-colouring of $G$ there is a rainbow $H$ in $G$. It is known that, for every graph $H$, an asymptotic upper bound for the threshold function $p_H^{\mathrm{rb}}=p_H^{\mathrm{rb}}(n)$ of this property for the random graph $G(n,p)$ is $n^{-1/m^{(2)}(H)}$, where $m^{(2)}(H)$ denotes the so-called maximum $2$-density of $H$. Extending a result of Nenadov, Person, Škorić, and Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower bound for $p_{K_k}^{\mathrm{rb}}$ for $k\ge 5$. Furthermore, we show that $p_{K_4}^{\mathrm{rb}}=n^{-7/15}$.

16 pages