
Given two vertex-ordered graphs G and H, the ordered Ramsey number R_<(G,H) is the smallest N such that whenever the edges of a vertex-ordered complete graph K_N are red/blue-colored, then there is a red (ordered) copy of G or a blue (ordered) copy of H. Let P_n^t denote the t-th power of a monotone path on n vertices. The ordered Ramsey numbers of powers of paths have been extensively studied. We prove that there exists an absolute constant C such that R_<(K_s,P_n^t)\leq R(K_s,K_t)^{C} \cdot n holds for all s,t,n, which is tight up to the value of C. As a corollary, we obtain that there is an absolute constant C such that R_<(K_n,P_n^t)\leq n^{Ct}. These results resolve a problem and a conjecture of Gishboliner, Jin and Sudakov. Furthermore, we show that R_<(P_n^t,P_n^t)\leq n^{4+o(1)} for any fixed t. This answers questions of Balko, Cibulka, Král and Kyncl, and of Gishboliner, Jin and Sudakov.
Joint work with Antonio Girao and Barnabas Janzer.