11 problems

Erdős–Hajnal conjecture. Every proper hereditary class of graphs has the Erdős–Hajnal property.

Multicolor Erdős–Hajnal conjecture. There exists $\varepsilon>0$ such that every coloring of the edges of $K_n$ contains either $k$ vertices whose edges are colored according to…

Linear-dimension strengthening. The multicoloured geometric Erdős–Hajnal conjecture and the geometric Erdős–Hajnal conjecture for induced restrictions should both hold with…

Geometric Erdős–Hajnal conjecture. For every prime power $q$ and all $k\in\mathbb N$ and $H\subseteq\operatorname{PG}(k-1,q)$, there exist $\delta,C>0$ such that, for all…

Multicoloured Erdős–Hajnal conjecture. For all $1\le s_0\le s$ and $k\ge1$, and every $s_0$-colouring $c_0$ of $K_k$, there exist $\delta,C>0$ such that, for all $n\in\mathbb N$ an…

Let $H$ be a tournament. An $H$-free tournament is one that does not contain $H$ as a not necessarily induced subdigraph. For a tournament $T$, write $\vec\alpha(T)$ for its maximu…

Size-version of the Erdős–Hajnal conjecture. There is a positive constant $\epsilon$ such that every colouring $c$ of $K_n$ in colours from $[s]$ avoiding $T$ satisfies

Forest characterization conjecture. If a tournament $H$ has an ordering of its vertices for which the backward arc digraph is a forest, then $H$ has the strong Erdős–Hajnal propert…

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 $H$ be a graph. For a graph $G$, let $\boldsymbol{\alpha}(G)$ be its maximum independent-set size and $\boldsymbol{\omega}(G)$ its maximum clique size. Pivot-minor Erdős–Hajnal…

Local-to-global tournament coloring conjecture. There is a function $f$ such that every $t$-local tournament $T$ satisfies