Geoffrey Exoo writes: I suggest using Cayley tournaments arising from nonabelian groups of order pq, where p and q are odd primes with q mod p = 1. They seem to work fairly well. Here's an example for the nonabelian group of order 111 = 3 * 37. There appears to be no acyclic subgraph of order 11? (Later note: Exoo rechecked and confirmed.) If so, that proves that R(11) > 111, a new record lower bound: 111x111 Adjacency matrix (please view on wide screen): 000110110101100001010111101000110000010001110010000111010111011111100001011101101100000001100111100001101101011 100001111111001000001001011101011011110001101000100110010000001110010011010101000101011100110011111111000100110 110100110110101100001010111101000110000010001110010000111010111011111100001011101101100000001100111100001101100 010011100000111100000110101001010000110001011010110101110111010001001010101111101010111100110011011110010000100 011000001111111001000001001011101011011110001101000100110010000001110010011010101000101011100110011111111000101 101010100110110101100001010111101000110000010001110010000111010111011111100001011101101100000001100111100001101 000010011100000111100000110101001010000110001011010110101110111010001001010101111101010111100110011011110010001 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