6 problems

Grzesik–Kielak conjecture. The balanced blow-up of $C_{\ell+2}$ asymptotically attains the maximum number of $k$-cycles that an $n$-vertex $C_{\ell}$-free graph can contain; equiva…

Let $T$ be a tree on $k$ vertices and let $r\geq3$. Write … For a graph $G$, let ${\mathcal N}(K_r,G)$ denote the number of copies of $K_r$ in $G$, and let ${\mathrm{ex}}(n,K_r,T)$…

Blow-up extremal-structure conjecture. There exists $n_0$ such that for all $n\geq n_0$, if $G$ is a $C_{\ell}$-free graph on $n$ vertices maximising the number of $k$-cycles, then…

Erdős's conjecture. The number of pentagons in any triangle-free graph on $n$ vertices is at most $(n/5)^5$.

Let $K_{1,s}$ and $K_{a,b}$ be complete bipartite graphs with $b\leq a<s$, and let $m\gg n^{1+s-1}$. Define $q$ and $r$ by … Here $K_q^*$ and $\overline{K_r^*}$ denote the correspo…

Let $K_{1,s}$ and $K_{a,b}$ be complete bipartite graphs with $b\leq a<s$, and let $\operatorname{satex}(n,K_{1,s}:m,K_{a,b})$ denote the minimum number of copies of $K_{a,b}$ in a…