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Let  U  5  be the tournament with vertices  v  1 , …,  v  5  such thatv  2   →   v  1    , andv  i   →   v  jif    j  −  i  ≡  1   ,    2    (  mod    5  )    and    {  i  ,  j  }  ≠  {  1  ,  2  }   . In this article, we describe the tournaments that do not have  U  5  as a subtournament. Specifically, we show that if a tournament  G  is “prime”—that is, if there is no subset    X  ⊆  V  (  G  )   ,    1  <  |  X  |  <  |  V  (  G  )  |   , such that for all    v  ∈  V  (  G  )    ∖    X   , either    v  →  x    for all    x  ∈  X    or    x  →  v    for all    x  ∈  X   —then  G  is  U  5 ‐free if and only if either  G  is a specific tournament    T  n    or    V  (  G  )    can be partitioned into sets  X ,  Y ,  Z  such that    X  ∪  Y   ,    Y  ∪  Z   , and    Z  ∪  X    are transitive. From the prime  U  5 ‐free tournaments we can construct all the  U  5 ‐free tournaments. We use the theorem to show that every  U  5 ‐free tournament with  n  vertices has a transitive subtournament with at least    n    log  3   2vertices, and that this bound is tight.

Structure Theorem for U 5 ‐free Tournaments