Directed graphs with lower orientation Ramsey thresholds

Abstract

We investigate the threshold [[EQUATION]] for the Ramsey-type property [[EQUATION]], where [[EQUATION]] is the binomial random graph and [[EQUATION]] indicates that every orientation of the graph [[EQUATION]] contains the oriented graph [[EQUATION]] as a subdigraph.  Similarly to the classical Ramsey setting, the upper bound [[EQUATION]] is known to hold for some constant [[EQUATION]], where [[EQUATION]] denotes the maximum 2-density of the underlying graph [[EQUATION]] of [[EQUATION]].  While this upper bound is indeed the threshold for some [[EQUATION]], 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 [[EQUATION]]-free graphs) and arbitrarily rooted transitive triangles.