5 problems

Let $H$ be a graph. For $\rho>0$, a set $S\subseteq V(G)$ is weakly $\rho$-restricted if one of $G[S]$ and $\overline{G}[S]$ contains at most $\rho|S|^2$ edges. Fox–Sudakov polynom…

Let $H$ be a graph. Say that $H$ has the Erdős–Hajnal property if it satisfies the Erdős–Hajnal conjecture, and say that it has the strong polynomial Rödl property in the sense use…

Let $H$ be a tournament. It has the strong Erdős–Hajnal property if there exists $epsilon_H>0$ such that every $H$-free tournament $T$ contains disjoint vertex subsets $A$ and $B$…

A class of tournaments $\mathcal T$ has the $BIG\Rightarrow BIG$ property if there exists a function $f$ such that, for every $T\in\mathcal T$, if…

Strong Erdős–Hajnal conjecture for adjacency sketchable classes. All adjacency sketchable graph classes have the strong Erdős–Hajnal property.