We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighbourhood is as large as its first outneighbourhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament of order 2n − 2 contains every arbor...

We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that in the (m : b) clique game played on KN , the complete graph on N vertices, Maker can achieve a Kq for q = ( m log 2 (b+1) − o(1) ) · logN , which partially solves an open problem by Beck. Moreover, we show that in the (1:1) clique game played on KN for a sufficiently large N , Make...

We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvátal and Erdős. We show that in the (m : b) game played on KN , the complete graph on N vertices, Maker can achieve a Kq for q = ( m log 2 (b+1) − o(1) ) · log2 N , which partially solves an open problem by Beck. Moreover, we show that in the (1 : 1) game played on KN for a sufficiently large N , Maker can achi...

Recent experimental results indicate that women do not like competitive environments as much as men. This paper presents an experimental design giving participants the opportunity to enter a tournament as part of a team rather than alone. While a large and significant gender gap in entry in the individual tournament is found in line with the literature, no gender gap is found in entry in the te...

A version of the Erdős-Hajnal conjecture for tournaments states that for every tournament H every tournament T that does not contain H as a subtournament, contains a transitive subtournament of size at least n (H) for some (H) > 0, where n is the order of T . For any fixed tournament H we can denote by n0(H) the supremum over all ≥ 0 satisfying the following statement: every tournament T of ord...

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...