For two graphs $G$ and $H$, write $G stackrel{mathrm{rbw}}{longrightarrow} H$ if $G$ has the property that every {sl proper} colouring of its edges yields a {sl rainbow} copy of $H$. We study the thresholds for such so-called {sl 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 $n^{-1/m_2(K_{leftlceil s/2 rightrceil})}$; 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. In this paper, we also consider the property $G cup mathbb{G}(n,p) stackrel{mathrm{rbw}}{longrightarrow} C_{2ell - 1}$, and show that the threshold for this property is $n^{-2}$ for every $ell geq 2$; in particular, it does not depend on the length of the cycle $C_{2ell - 1}$. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.
Download