Oriented Graphs with Lower Orientation Ramsey Thresholds

Abstract

For a graph G and an oriented graph \({\vec {H}}\), let \(G\rightarrow {\vec {H}}\) denote the property that every orientation of G contains a copy of \({\vec {H}}\). We investigate the threshold \(p_{{\vec {H}}}=p_{{\vec {H}}}(n)\) for \(G(n,p)\rightarrow {\vec {H}}\), where G(n, p) is the binomial random graph. Similarly to the classical (edge-colouring) Ramsey setting, \(p_{{\vec {H}}}\leqslant n^{-1/m_2({\vec {H}})}\), where \(m_2({\vec {H}})\) denotes the maximum 2-density of \({\vec {H}}\). While \(n^{-1/m_2({\vec {H}})}\) gives the correct order of magnitude for acyclic orientations of cycles and complete graphs with at least 4 vertices, this is known not always to be the case. We extend the examples in that category, describing a large family of oriented graphs \({\vec {H}}\) such that \(p_{{\vec {H}}}\ll n^{-1/m_2({\vec {H}})}\).