10 problems

Lovász–Simonovits clique density conjecture. For every $r\ge3$ and every graph $G$,

Let $\mathrm{er}_{s}(\{\overline{K}_3,C_5\})$ denote the minimum asymptotic density of $s$-cliques among graphs avoiding both $\overline{K}_3$ and $C_5$. Theorem's extension conjec…

Let $s\ge 1$, let $\varepsilon>0$, and let $G$ be the complement of a string graph. Here $\varrho_{4s+1}(G)$ denotes the density of copies of $K_{4s+1}$ in $G$, and $K_{s+1}[t]$ de…

Let $K_r$ denote the complete graph on $r$ vertices. For integers $n$ and $m$ with $0\leq m\leq {n\choose 2}$, let $G_r(n,m)$ be the minimum number of copies of $K_r$ in an $(n,m)$…

Lovász–Simonovits conjecture. The asymptotic lower bound on the density of $K_s$ is

Let $\mathcal{G}(\Delta,\omega)$ be the class of graphs with maximum degree at most $\Delta$ and clique number at most $\omega$, and let $f_t(\Delta,\omega)$ be the supremum of…

For $k\geq 2$, let $\widetilde{\operatorname{BT}}(k)$ be obtained from $T_2(2k)$ by deleting an edge and adjoining a new vertex to the two endpoints of that edge, and let … Let…

Lovász–Simonovits extremal graph conjecture. An extremal graph for $\mathcal K_k(n,M)$ is obtained from a complete $t$-partite graph, for an appropriate $t$, by adding a matching t…

Conjectured value of $c_{4,4}$.

Logarithmic balanced-clique conjecture. There is an absolute constant $c>0$ such that $G$ contains a copy of $K_{t,t,t}$ with