On avoidable and unavoidable claws

A claw of degree k is a directed tree consisting of k paths emerging from a common root. We prove that every claw of order n with degree less than 1950n appears in every n-vertex tournament. We also construct avoidable claws with degree approaching 1123n. Thus for large n, the maximum λ such that every claw with degree λn appears in every n-vertex tournament satisfies λ ⩽ 1123. This improves earlier bounds.