The anti-Ramsey threshold of complete graphs

TL;DR

This paper determines the precise threshold probabilities for the appearance of rainbow complete subgraphs in random graphs under proper edge-colorings, extending previous bounds and providing exact thresholds for certain cases.

Contribution

It establishes a matching lower bound for the rainbow $K_k$ threshold in random graphs for $k eq 4$, and computes the exact threshold for $K_4$, advancing understanding of rainbow subgraph thresholds.

Findings

Matching lower bounds for $p^{rb}_{K_k}$ when $k eq 4$

Exact threshold $p^{rb}_{K_4} = n^{-7/15}$

Extension of previous asymptotic bounds to precise thresholds

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 $p^{\rm rb}_H=p^{\rm rb}_H(n)$ 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, \v{S}kori\'c, and Steger [J. Combin. Theory Ser. B 124 (2017),1-38] we prove a matching lower bound for $p^{\rm rb}_{K_k}$ for $k\geq 5$. Furthermore, we show that $p^{\rm rb}_{K_4} = n^{-7/15}$.