11 problems

Edge-statistics conjecture. For every $\eps>0$, if $k$ is sufficiently large in terms of $\eps$ and $1\leq\ell\leq\frac12\binom{k}{2}$, then

The Edge-statistics Conjecture. For all integers $k$ and $\ell$ with $0<\ell<{k\choose 2}$,

For integers $h$ and $\ell$ with $0<\ell<\binom{h}{2}$, let $\mathcal Q_{h,\ell}$ be the set of all graphs on $h$ vertices having exactly $\ell$ edges, and let…

Let $\operatorname{ind}(k,\ell)$ be the edge-inducibility of $k$-vertex subsets inducing exactly $\ell$ edges in graphs. As $k\to\infty$, consider the values of $\ell$ in the range…

Sharp inducibility conjecture. For all such $k$, $\ell$, and $H$,

Asymmetric fractalizer conjecture. There exists an asymmetric fractalizer on at most $9$ vertices.

Let $F$ be a complete partite graph. The inducibility problem for $F$ asks for the maximum possible induced $F$-density in graphs of increasing order, and is perfectly stable when…

Choi–Lidický–Pfender conjecture.

Let $C_k$ denote the cycle on $k$ vertices. For a graph $H$, its induced density is the number of induced copies of $H$ divided by $\binom{|G|}{|H|}$; write $I(H;n)$ for the maximu…

Let $\vec P_3$ be the oriented path on three vertices and $\vec C_4$ the directed cycle on four vertices. For an oriented graph $H$, call an oriented graph extremal for $H$ if it a…

Let $\vec P_3$ be the oriented path on three vertices, and let $i(\vec P_3)$ denote its inducibility, the maximal induced density of $\vec P_3$ in an oriented graph. Thomassé's con…