Ordered Ramsey numbers (Q5915485)

Let \(TT_n\) denote the transitive tournament on \(n\) vertices and let \(D_1,D_2,\dots, D_k\) be acyclic digraphs. The ordered Ramsey number \(\rho(D_1,D_2,\dots, D_k)\) is the smallest integer \(n\) so that any \(k\)-coloring of the edges of \(TT_n\) admits, for some index \(i\), a copy of \(D_i\) with all of its edges colored with the \(i\)th color. The authors compute several infinite classes of these ordered Ramsey numbers.