On Tournaments and Their Largest Transitive Subtournaments

In this work we present some results in connection with the following problem, posed by Erdös and Moser: For each positive integer n, determine the greatest integer v(n) such that all tournaments of order n contain the transitive subtournament of order v(n) (denoted ???(?)). It is known that v(n) = 3 for 4 ≤ n ≤ 7, v(n) = 4 for 8 ≤ n ≤ 13, ?(?) = 5 for 14 ≤ n ≤ 27, and ?(n) ≥ 6 for n > 27. Moreover, the uniqueness of the tournaments of orders 7 and 13, free of TT4 and TT5, respectively, has been proved. Here we prove that there exists only one tournament of order 27 free of TT6 and that every tournament of order 55 contains TT7, which impliesv(n) ≥ ⌊log2(n/55⌋ + 7, forn ≥ 55, improving the best lower bound known for v(n).