Ordered Trees with Linear Off-Diagonal Ramsey Numbers

We characterize the following three classes of ordered trees: $$ (1) trees $T$ satisfying $R_{<}(T,K_n)=O(n)$; $$ (2) $\chi$-bounding trees; $$ (3) Ramsey-good trees. In particular, we show that the first two classes coincide. The second characterization answers a question of Axenovich, Rollin and Ueckerdt, and the third resolves a problem of Balko and Poljak. In our proof, we also introduce several novel constructions of ordered graphs with sublinear independence number, giving lower bounds on $R_{<}(F,K_n)$ for various ordered forests $F$.