New lower bounds of fifteen classical Ramsey numbers

An algorithm to compute lower bounds for the classical Ramsey numbers R(Ql' Q2," " Qn) is developed. The decompositions of each of fifteen complete graphs of prime order into n circulant graphs are constructed and their properties are studied by computer search. This leads to new lower bounds of eight multicolor and seven 2-color Ramsey numbers, namely, R(3, 3, 6) 2: 54, R(3, 3, 7) 2: 72, R(3, 3, 9) 2: 1l0, R(3, 3, ll) 2: 138, R(3, 3, 3, 5) 2: 1l0, R(3, 3, 3, 6) 2: 138, R(3, 3, 3, 7) 2: 194, R(3, 3, 4, 4) ~ 114 and R( 4,19) 2: 194, R( 4,20) 2: 230, R( 4,21) 2: 242, R(5, 17) 2: 282, R(5, 19) 2: 338, R(6, 17) 2: 500, and R(7, 17) 2: 548. *Research supported by Guangxi Natural Science Foundation tVisiting professor in Guangxi Computing Center Australasian Journal of Combinatorics 19(1999), pp.91-99 The classical Ramsey number R(ql, q2,' ", qn) (for n 2: 2) is the smallest integer r such that if the edges of Kr, the complete graph of order r, are colored with n colors, then for some i E {I, 2, ... ,n} there is a monochromatic K q;. Only nine exact values of 2-color Ramsey numbers have been found up to now. The number of best lower bounds is no more than 40 (see [11)). Prior to our results, no non-trivial lower bounds for R(ql, q2) when ql 2: 4 and q2 ~ 16 were known. The following are two of the results we obtained previously: R( 4,12) 2: 128 [13] R(6, 12) ~ 224 [7]. When n 2: 3, less is known about R(ql, q2," " qn). So far, only one exact value of a multicolor Ramsey number has been found: R(3,3,3) = 17 [4]. Some non-trivial lower and upper bounds have been found, mostly by computer, for example: 30 ~ R(3, 3, 4) :::; 31 45 ~ R(3,3,5):::; 57 55 ~ R(3, 4, 4) :::; 79 87 ~ R(3, 3, 3, 4) :::; 155 80 ~ R(3, 4, 5) :::; 161 We ourselves obtained the following: [5,10] [1,6,2] [6,2] [2,3] [2,3]. 458 :::; R( 4,4,4,4) [12] 90 :::; R(3, 3, 9) [8] 108 :::; R(3, 3,11) [9]. For details, please refer to the authoritative dynamic survey [11], which also provides many references. In this paper, we present a new algorithm to compute bounds on classical Ramsey numbers. The algorithm is based on circulant graphs of prime order. It is so efficient that the following fifteen new lower bounds are obtained, of which eight are for multicolor Ramsey numbers and the other seven (which have been quoted by Radziszowski [11]) are for 2-color ones. Theorem 1 R(3, 3, 6) ~ 54, R(3, 3, 7) ~ 72, R(3, 3, 9) 2: 110, R(3, 3, 11) ~ 138, R(3, 3, 3, 5) 2: 110, R(3, 3, 3, 6) ~ 138, R(3, 3, 3, 7) 2: 194, R(3, 3, 4, 4) ~ 114; R(4,19) ~ 194, R(4,20) 2: 230, R(4,21) ~ 242, R(5,17) ~ 282, R(5,19) 2: 338, R(6, 17) ~ 500, R(7, 17) 2: 548. 1 Circulant graphs of prime order For a given prime integer p = 2m+l, let Zp {-m,···, -1,0,1, ... , m} = [-m, m]. (For s ~ t, we denote the set {8, 8 + 1"", t} by [8, t].) In the rest of the paper, unless the contrary is implied, all integers will be considered to be members of Zp,