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By Shi M., Yuan X., Cai M.

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The transitive tournament Tn has index one for all n. In view of Theorem 20, it remains only to exhibit a tournament Tn with index two for n ≥ 16. If n ≥ 16, let n = 3h + r, where r = 16, 17, or 18, and h is an integer. Let Tn be the tournament that can be decomposed in the manner described earlier into the subtournaments T (i) (i = 1, 2, . . , h + 2), where T (i) is the transitive tournament Tr−7 , T (2) is the tournament T7 in which pi → pj if and only if j − i is a quadratic residue modulo seven, and the tournament T (i) is a 3-cycle, for i = 3, 4, .

Corollary. 2. Almost all tournaments have diameter three. Notice that, if the definition of the diameter of a tournament involved ordered pairs of distinct nodes only, then we could assert that almost all tournaments have diameter two. Exercises 1. Supply the details omitted in the proof of Theorem 18. 2. Prove that d ≤ n − 1 for any strong tournament Tn with at least four nodes. 3. Prove that d ≤ max{3, sn − s1 + 2}, where sn and s1 denote the largest and smallest score in a strong tournament Tn .

Choose the integer r so that g(r) ≤φ+ , r for an arbitrary positive value of . For any given value of n, let n = rs + t, where 0 < t ≤ r. Then g(rs + t) g(rs) g(t) g(n) = ≤ + n n n n sg(r) g(t) rs g(t) ≤ + ≤ (φ + ) + n n n n 1 ≤ φ + + max{g(1), . . , g(r)}. n Therefore, g(n) ≤φ+ n for every positive . The lemma follows immediately. In forming a spanning path of a tournament Tn , we may choose the first k nodes of this path, form a path spanning these k nodes, and then join this path to a path spanning the remaining n − k nodes.

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