Graphs with many r-cliques have large complete r-partite subgraphs

For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erd\H{o}s--Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lov\'{a}sz and Simonovits from the 1970s states that, for every $r\ge 3$, if $n$ is sufficiently large then, for every $e\le \binom{n}{2}$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part. In thi...

Emergence of dominating cliques in Erd\"os-R\'enyi random graph model ${\bbbg(n,p)}$ is investigated in this paper. It is shown this phenomenon possesses a phase transition. Namely, we have argued that, given a constant probability $p$, an $n$-node random graph $G$ from ${\bbbg(n,p)}$ and for $r= c \log_{1/p} n$ with $1 \leq c \leq 2$, it holds: (1) if $p > 1/2$ then an $r$-node clique is dominating in $G$ almost surely and, (2) if $p \leq (3 - \sqrt{5})/2$ then an $r$-node c...

Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn, Lo and Osthus leads to the best known minimum degree thresholds for exact (non-fractional) $F$-decompositions for a wide class of graphs~$F$ (including large cliques). For general $k$-uniform hypergraphs, we give a short argument which shows that there ex...

It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$ cliques suffice; and also prove that, even for graphs with no stable set of size four, we may need more than linearly many cliques. This settles two questions discussed at a recent conference in Lyon.

Let $\mathbf{k} := (k_1,\dots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been s...

We investigate lower bounds on the average degree in r-cliques in graphs of order n and size greater than t(r,n), where t(r,n) is the size of the Turan graph on n vertices and r color classes. Continuing earlier research of Edwards and Faudree, we completely prove a conjecture of Bollobas and Erdoes from 1975.

We study how many copies of a graph $F$ that another graph $G$ with a given number of cliques is guaranteed to have. For example, one of our main results states that for all $t\ge 2$, if $G$ is an $n$ vertex graph with $kn^{3/2}$ triangles and $k$ is sufficiently large in terms of $t$, then $G$ contains at least \[\Omega(\min\{k^t n^{3/2},k^{\frac{2t^2}{3t-1}}n^{\frac{5t-2}{3t-1}}\})\] copies of $K_{2,t}$, and furthermore, we show these bounds are essentially best-possibl...

A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions show...

We show that for every two cycles $C,D$, there exists $c>0$ such that if $G$ is both $C$-free and $\overline{D}$-free then $G$ has a clique or stable set of size at least $|G|^c$. (``$H$-free'' means with no induced subgraph isomorphic to $H$, and $\overline{D}$ denotes the complement graph of $D$.) Since the five-vertex cycle $C_5$ is isomorphic to its complement, this extends the earlier result that $C_5$ satisfies the Erd\H{o}s-Hajnal conjecture. It also unifies and streng...

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then $\omega_{\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\alpha(\gamma,p)=\gamma\log\frac{\gamma}{p}+(1-\gamma)\log\frac{1-\gamma}{1-p}$, we show that $\omega_{\gamma}(G_{n,p})$ is one of the two integers closest t...