An approximate version of Sumner's universal tournament conjecture

Authors:
Daniela KüHn;Richard Mycroft;Deryk Osthus
Affiliations:
School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom;School of Mathematical Sciences, Queen Mary, University of London, London, E1 4NS, United Kingdom;School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom
Venue:
Journal of Combinatorial Theory Series B
Year:
2011

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Abstract

Sumner@?s universal tournament conjecture states that any tournament on 2n-2 vertices contains a copy of any directed tree on n vertices. We prove an asymptotic version of this conjecture, namely that any tournament on (2+o(1))n vertices contains a copy of any directed tree on n vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed @D, any tournament on (1+o(1))n vertices contains a copy of any directed tree on n vertices with maximum degree at most @D.