Claims

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1963_10_01_erdos: Erdős (Math. Gaz. 1963) shows that tournaments with Schütte's property S_k exist, finds f(1) = 3 and f(2) = 7, and proves 2^(k+1) − 1 <= f(k) and f(k) <= 2^k k^2 log(2+ε) for large k; refereed.

1965_10_01_szekeres_szekeres: E. and G. Szekeres (Math. Gaz. 1965) prove f(k) >= (k+2)2^(k−1) − 1 for the least order of a tournament with Schütte's property S_k, and f(3) = 19; refereed, the best known lower bound for general k.

2004_03_01_reid_mcrae_hedetniemi_hedetniemi: Reid, McRae, Hedetniemi and Hedetniemi (Australas. J. Combin. 2004) show that a tournament of domination number at least 5 has at least 48 vertices, so f(4) >= 48, and prove f(3) = 19; refereed.

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