On a problem of Erdős and Rado (Q1387613)

Improved upper and lower bounds for the digraph Ramsey numbers \(r(K_n^*,L_m)\) are verified, where \(r(K_n^*,L_m)\) is the smallest integer \(p\) such that any digraph of order \(p\) contains either an independent set with \(n\) vertices or a transitive tournament of order \(m\). For example, it is shown that \(r(K_n^*, L_3) \leq n^2\), and in particular that \(13 < r(K_4, L_3) \leq 16\). Recursive upper bounds are given that improve on previous bounds of Erdős and Rado, and also of Bermond.