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 19/50n appears in every n-vertex tournament. We also construct avoidable claws with degree approaching 11/23n. Thus for large n, the maximum λ such that every claw with degree λn appears in every n-vertex tournament satisfies λ ≤ 11/23. This improves earlier bounds. © 1998 Elsevier Science B.V. All rights reserved.