Large Rainbow Cliques in Randomly Perturbed Dense Graphs

Abstract

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every proper coloring of its edges yields a rainbow copy of $H$. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form $G \cup \mathbb{G}(n,p)$, where $G$ is an $n$-vertex graph with edge-density at least $d$, and $d$ is a constant that does not depend on $n$. Our results in this paper, combined with our results in a companion paper, determine the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_s$ for every $s$. In this paper, we show that for $s \geq 9$ the threshold is $n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}$; in fact, our $1$-statement is a supersaturation result. This turns out to (almost) be the threshold for $s=8$ as well, but for every $4 \leq s \leq 7$, the threshold is lower; see our companion paper for more details. Also in this paper, we determine that the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}$ is $n^{-2}$ for every $\ell \geq 2$; in particular, the threshold does not depend on the length of the cycle $C_{2\ell - 1}$. For even cycles, and in fact any fixed bipartite graph, no random edges are needed at all; that is, $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ always holds, whenever $G$ is as above and $H$ is bipartite.