On Ordered Ramsey Numbers of Matchings Versus Triangles

For graphs G^< and H^< with linearly ordered vertex sets, the ordered Ramsey number r_<(G^<,H^<) is the smallest positive integer N such that any red-blue coloring of the edges of the complete ordered graph K^<N on N vertices contains either a blue copy of G^< or a red copy of H^<. Motivated by a problem of Conlon, Fox, Lee, and Sudakov (2017), we study the numbers r_<(M^<,K^<₃) where M^< is an ordered matching on n vertices. We prove that almost all n-vertex ordered matchings M^< with interval chromatic number 2 satisfy r_<(M^<,K^<₃) ∈ Ω((n/log n)5/4) and r_<(M^<,K^<₃) ∈ O(n7/4), improving a recent result by Rohatgi (2019). We also show that there are n-vertex ordered matchings M^< with interval chromatic number at least 3 satisfying r_<(M^<,K^<₃) ∈ Ω((n/log n)4/3), which asymptotically matches the best known lower bound on these off-diagonal ordered Ramsey numbers for general n-vertex ordered matchings.