On 1-subdivisions of transitive tournaments

Abstract

The oriented Ramsey number (r) over right arrow (H) for an acyclic digraph H is the minimum integer n such that any n-vertex tournament contains a copy of H as a subgraph. We prove that the 1-subdivision of the k-vertex transitive tournament Hk satisfies (r) over right arrow (H-k) = O(k(2) log log k). This is tight up to multiplicative log log k-term. We also show that if T is an n-vertex tournament with Delta+(T) - delta(+)(T) = O(n/k) - k(2), then T contains a 1-subdivision of (K) over right arrow (k), a complete k-vertex digraph with all possible k(k - 1) arcs. This is tight up to multiplicative constant.