The anti-Ramsey threshold of complete graphs

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 $p^{\rm rb}_{K_4} = n^{-7/15}$.Comment: 19 page

Large rainbow cliques in randomly perturbed dense graphs

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 $n^{-1/m_2(K_{\left\lceil s/2 \right\rceil})}$; 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. In this paper, we also consider the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} C_{2\ell - 1}$, 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 $C_{2\ell - 1}$. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all.Comment: 21 pages; some typos fixed in the last versio

Graph removal lemmas

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.Comment: 35 page

Combinatorics, Probability and Computing

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. The workshop also emphasized connections between probabilistic combinatorics and discrete probability

Small rainbow cliques in randomly perturbed dense graphs

For two graphs $G$ and $H$, write $G \stackrel{\mathrm{rbw}}{\longrightarrow} H$ if $G$ has the property that every \emph{proper} colouring of its edges yields a \emph{rainbow} copy of $H$. We study the thresholds for such so-called \emph{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 $d >0$, and $d$ is independent of $n$. In a companion article, we proved that the threshold for the property $G \cup \mathbb{G}(n,p) \stackrel{\mathrm{rbw}}{\longrightarrow} K_\ell$ is $n^{-1/m_2(K_{\left\lceil \ell/2 \right\rceil})}$, whenever $\ell \geq 9$. For smaller $\ell$, the thresholds behave more erratically, and for $4 \le \ell \le 7$ they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for \emph{large} cliques. In particular, we show that the thresholds for $\ell \in \{4, 5, 7\}$ are $n^{-5/4}$, $n^{-1}$, and $n^{-7/15}$, respectively. For $\ell \in \{6, 8\}$ we determine the threshold up to a $(1 + o(1))$-factor in the exponent: they are $n^{-(2/3 + o(1))}$ and $n^{-(2/5 + o(1))}$, respectively. For $\ell = 3$, the threshold is $n^{-2}$; this follows from a more general result about odd cycles in our companion paper.Comment: 37 pages, several figures; update following reviewer(s) comment

Small rainbow cliques in randomly perturbed dense graphs

For two graphs G and H, write G rbw −→ H if G has the property that every proper colouring 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 ∪ G(n, p), where G is an n-vertex graph with edge-density at least d > 0, and d is independent of n. In a companion paper, we proved that the threshold for the property G ∪ G(n, p) rbw −→ K` is n −1/m2(Kd`/2e) , whenever ` ≥ 9. For smaller `, the thresholds behave more erratically, and for 4 ≤ ` ≤ 7 they deviate downwards significantly from the aforementioned aesthetic form capturing the thresholds for large cliques. In particular, we show that the thresholds for ` ∈ {4, 5, 7} are n −5/4 , n −1 , and n −7/15, respectively. For ` ∈ {6, 8} we determine the threshold up to a (1 + o(1))-factor in the exponent: they are n −(2/3+o(1)) and n −(2/5+o(1)), respectively. For ` = 3, the threshold is n −2 ; this follows from a more general result about odd cycles in our companion paper