Abstract.
Erdös and Moser posed the problem of determining, for each integer n>0, the greatest integer v(n) such that all tournaments of order n contain the transitive subtournament of order v(n) (denoted TT v(n)). It is known that v(n)=3 for , v(n)=4 for , v(n)=5 for , and for n>27. Moreover, the uniqueness of the tournaments free of TT 4 of orders 6 and 7, and free of TT 5 and TT 6 of orders 13 and 27, respectively, has been established. Here we prove that the tournaments of orders 12 and 26, free of TT 5 and TT 6, respectively, are also unique. Then, we see that all tournaments of order 54 contain TT 7 (improving the best lower bound known for v(n)). Finally, with the aid of a computer, we obtain the orders cv(r) and gv(s) of the biggest transitive tournaments contained, respectively, in all circulant tournaments of order r≤55 and in each Galois tournament of order s<1000, i.e., in the tournament with set of vertices the Galois field of order s (whenever it exists) and edge directions induced by the quadratic residues. We get better upper bounds of v(n), for n≤991.
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Received: March 25, 1996 / Revised: September 25, 1996
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Sanchez-Flores, A. On Tournaments Free of Large Transitive Subtournaments. Graphs Comb 14, 181–200 (1998). https://doi.org/10.1007/s003730050025
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DOI: https://doi.org/10.1007/s003730050025