Structure theorem for U5-free tournaments
Gaku Liu

TL;DR
This paper characterizes prime tournaments that do not contain a specific 5-vertex subtournament $U_5$, providing a structural theorem and implications for transitive subtournaments.
Contribution
It offers a complete structural description of $U_5$-free tournaments, including a construction method and bounds on transitive subtournaments.
Findings
Prime $U_5$-free tournaments are either a specific tournament $T_n$ or have a tripartition with transitive unions.
All $U_5$-free tournaments can be generated from prime $U_5$-free tournaments.
Every $U_5$-free tournament with $n$ vertices has a transitive subtournament of size at least $n^{rac{ ext{log}_3 2}$}.
Abstract
Let be the tournament with vertices , ..., such that , and if , and . In this paper we describe the tournaments which do not have as a subtournament. Specifically, we show that if a tournament is "prime"---that is, if there is no subset , , such that for all , either for all or for all ---then is -free if and only if either is a specific tournament or can be partitioned into sets , , such that , , and are transitive. From the prime -free tournaments we can construct all the -free tournaments. We use the theorem to show that every -free tournament with vertices has a…
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