7 problems

Let $\mathcal{C}$ be a sparse hereditary class of graphs with unbounded tree-width. A hereditary graph class is minimal of unbounded tree-width if it has unbounded tree-width, whil…

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

Let $\mathcal{C}$ be a hereditary class of finite graphs. The bounded clique-width conjecture. If $\mathcal{C}$ is $2$-well-quasi-ordered, then $\mathcal{C}$ has bounded clique-wid…

Esperet's conjecture. Every chi-bounded hereditary class is poly-chi-bounded.

Let $\mu$ be a finite relational signature, and let $\mathscr S_{\mu}$ be the class of all relational structures of signature $\mu$ without any finite monomorphic decomposition. A…

Small conjecture. Every small hereditary class has bounded twin-width.

Let $\mathfrak D$ be a hereditary class of indecomposable ordered binary structures. Suppose that $\mathfrak D$ is hereditary well-quasi-ordered and hereditary algebraic. Sum-closu…