10 problems

Let $H_0$ be a tripartite $3$-graph one of whose parts is a singleton. For every $\varepsilon>0$, there exists $\delta>0$ such that, whenever $H$ is a $3$-partite $3$-graph on part…

Let $G$ be an $n$-vertex $r$-graph, let $s(G;m)$ denote the number of values of $f$ for which $G$ contains $m$ vertices spanning exactly $f$ edges, and let $g_r(m)$ be the threshol…

Directed Erdős–Hajnal conjecture. For every tournament $H$, there exists $\epsilon(H)>0$ such that every $H$-free tournament with $n$ vertices contains a transitive subtournament of…

Let $T$ be a tournament. A nebula is a tournament admitting an ordering whose vertices are partitioned into the vertex sets of stars and singleton components, with no restriction o…

Erdős–Hajnal conjecture. There are constants $A$ and $k$ such that

Positive EH-coefficient conjecture. Every tournament has a positive EH-coefficient.

Tournament Erdős–Hajnal conjecture. For every tournament $S$, there exists a constant $\delta(S)>0$ such that every $S$-free tournament $T$ satisfies

Self-complementary Erdős–Hajnal conjecture. For every graph $H$, there exists a constant $\epsilon(H)>0$ such that every $\{H,H^c\}$-free graph $G$ has either a clique or a stable…

Narrowness conjecture. For every graph $H$, there exists a constant $\alpha(H)\geq 1$ such that every $H$-free graph $G$ is $\alpha(H)$-narrow.

Perfect-subgraph formulation. For every graph $H$, there exists a constant $\psi(H) > 0$, such that every $H$-free graph $G$ has a perfect induced subgraph with at least…