0,3

A Ramsey-like number but defined for tournaments (i.e. directed graphs in which each node-pair is joined by exactly one arc) rather than undirected graphs.

It is not hard to show a(n) always exists and a(n) is nondecreasing.

The lower bounds a(4)>=8 and a(5)>=14 and a(6)>=28 arise from the cyclic tournaments with offsets 1,2,4 mod 7, ditto with offsets 1,3,9,2,6,5 mod 13, and the "QRgraph" in GF(3^3) with 27 vertices.

The following lower bounds a(n)>=P+1 arise from QRgraph(P) where P is prime and P=3 (mod 4): a(8)>=48, a(9)>=84, a(10)>=108, a(12)>=200, a(13)>=272.

This is almost certainly different from the other sequences currently in the OEIS which begin 1,2,4,8,14,28.

K. B. Reid, Tournaments, in Handbook of Graph Theory; see p. 167.

W. D. Smith, Partial Answer to Puzzle #21: Getting rid of cycles in directed graphs

Yahoo Groups, Range Voting

W. D. Smith, Survey on directed graph Ramsey Numbers.

Cf. A122027, A003141.

Sequence in context: A048238 A048140 A065616 this_sequence A118034 A096590 A068912

Adjacent sequences: A122023 A122024 A122025 this_sequence A122027 A122028 A122029

nonn

Warren D. Smith, warren.wds(AT)gmail.com, Sep 11 2006