Directed graphs with lower orientation Ramsey thresholds

\nWe investigate the threshold p⃗H = p⃗H (n) for the Ramsey-type property G(n, p) → ⃗H, where G(n, p) is the binomial random graph and G → ⃗H indicates that every orientation of the graph G contains the oriented graph ⃗ H as a subdigraph. Similarly to the classical Ramsey setting, the upper bound p⃗H ⩽ Cn−1/m2(⃗ H) is known to hold for some constant C = C( ⃗ H), where m2(⃗H) denotes the maximum 2-density of the underlying graph H of ⃗ H. While this upper bound is indeed the threshold for some ⃗H, this is not always the case. We obtain examples arising from rooted products of orientations of sparse graphs (such as forests, cycles and, more generally, subcubic {K3, K3,3}-free graphs) and arbitrarily rooted transitive triangles.\n