Linear Ramsey numbers and bounded co-chromatic number for finitely defined hereditary classes

Let $X$ be a hereditary class of graphs, meaning that it is closed under taking induced subgraphs, and suppose that $X$ is finitely defined, meaning that it has finitely many minimal forbidden induced subgraphs. The co-chromatic number of a graph $G$ is the minimum number of subsets into which $V(G)$ can be partitioned, with each subset either a clique or an independent set. Ramsey numbers are linear in $X$ if there is a constant $k$ such that

$$R_X(p,q)\leq k(p+q)$$

for all natural numbers $p$ and $q$.

Linear Ramsey–co-chromatic conjecture. A finitely defined hereditary class is of linear Ramsey numbers if and only if it has bounded co-chromatic number.

Classes of bounded co-chromatic number are known to have linear Ramsey numbers, but the converse fails for general hereditary classes. The conjecture asserts that the two notions coincide among finitely defined classes.