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

We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the the r-partite Turan graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.

In 1987, Kolaitis, Pr\"omel and Rothschild proved that, for every fixed $r \in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \leqslant (\log n)^{1/4}$. The proof combines a new (close to sharp) supersaturation version of the Erd\H{o}s-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollob\'as and Simonovits.

We estimate the maximum possible number of cliques of size $r$ in an $n$-vertex graph free of a fixed complete $r$-partite graph $K_{s_1, s_2, \ldots, s_r}$. By viewing every $r$-clique as a hyperedge, the upper bound on the Tur\'an number of the complete $r$-partite hypergraphs gives the upper bound $O\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. We improve this to $o\left(n^{r - {1}/{\prod_{i=1}^{r-1}s_i}}\right)$. The main tool in our proof is the graph removal lemma. ...

We show that the expected number of cliques in the Erd\H{o}s-R\'enyi random graph $G(n,p)$ is $n^{\frac1{-2\log p}(\log n-2\log\log n+O(1))}$.

We determine how large r-partite graphs can be found in r-uniform graphs with n vertices and Cn^r edges, where C is a slowly decreasing function of n. This refines results of Erdos from 1964.

Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, \ldots, A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$. Let $\cp(n,r)$ denote the minimum size of a clique partition of ${[n] \choose r}$. A classical theorem of de Bruijn and Erd\H os states that $\cp(n, 2) = n$. In this paper we study $\cp(n,r)$, and show in general that for each fixed $r \geq 3$, \[ \cp(n,r) \ge...

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs. The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\gamma n^2$ edges has been intensively studied in the liter...

The clique cover number of a graph G is the minimum number of cliques required to cover the edges of graph G. In this paper we consider the random graph G(n,p), for p constant. We prove that with probability 1-o(1), the clique number of G(n,p) is Theta(n^2/\log^2n).

In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.

Let $G$ be a $2$-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} - o(1))\log n}$ monochromatic complete subgraphs of size $r$, where \[ 0.3\log n < r < 0.7\log n. \] The previously known lower bound on the total number of monochromatic complete subgraphs, due to Sz\'{e}kely was $n^{0.1576\log n}$. We also prove that $G$ contains at least $n^{\frac{1}{7} \log n} $ monochromatic complete subgraphs of...