SOME INTERMEDIATE DATA FOR COMPUTATIONS OF R(4,4;3)=13 The First Classical Ramsey Number for Hypergraphs Is Computed B.D. McKay and S.P. Radziszowski In all tables t < C(n,3) Table I: |R(4,f,t)|, TR(4)=0, U(4)=1. f | 0 1 | sum t | | ---------------- 1 | 1 | 1 2 | 1 | 1 ---------------- sum | 1 1 | 2 Table II: |R(5,f,t)|, TR(5)=1, U(5)=5. f | 1 3 5 | sum t | | ------------------- 3 | 1 | 1 4 | 1 2 | 3 5 | 1 2 1 | 4 ------------------- sum | 3 4 1 | 8 Table III: |R(6,f,t)|, TR(6)=3, U(6)=15. f | 3 4 5 6 7 8 9 10 11 12 13 14 15 | sum t | | -------------------------------------------------- 6 | 1 | 1 7 | 3 2 | 5 8 | 4 9 7 2 | 22 9 | 15 15 17 2 1 | 50 10 | 7 20 25 15 2 1 | 70 -------------------------------------------------- sum | 12 18 29 17 32 17 17 2 2 1 0 0 1 | 148 Table IV: |R(7,f,t)|, TR(7)=7, U(7)=21. f | 7 9 11 13 15 17 19 21 | sum t | | --------------------------------------------------- 12 | 1 | 1 13 | 9 14 3 | 26 14 | 1 51 129 115 37 4 1 | 338 15 | 2 144 469 618 400 148 10 2 | 1793 16 | 5 281 1011 1520 1360 697 167 14 | 5055 17 | 8 389 1468 2277 2324 1354 436 61 | 8317 --------------------------------------------------- sum | 16 875 3091 4533 4121 2203 613 78 |15530 Table V. |TR(n,f)|: f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | n | | ----------------------------------------------------------------------- 5 | 1 1 1 | 6 | 1 1 2 1 2 2 3 1 1 1 1 | 7 | 1 5 18 37 71 94 91 42 | 8 | 1 4 11 38 127 321 | ----------------------------------------------------------------------- TR(7,7) - complement of the Steiner system S(7,3,2) TR(8,14) - the Steiner system S(8,4,3) It can be proved by hand that TR(8,15) = TR(8,16) = empty In general F in TR(n) has many nonisomorphic colorings in R(n), i.e. there are many systems G in R(n) such that Fe(G)=F Table VI. |TR(12,f)|, |R(12,f)| for f<135: f | 126 127 128 129 130 131 132 133 134 ------------------------------------------------- |R(12,f)| | 1 10 0 8 21 19 15 86 42 |TR(12,f)| | 1 1 0 3 8 6 14 39 35 Theorem 6: R(4,4;3)=13 Produced more than 200,000 nonisomorphic R(12,f,t) systems f ranging from TR(12)=126 to U(12)=159 t ranging from 104 to 116 997 self-complementary systems with t=110 Approximately 6E13 machine instructions carried out on different computers in Canberra and Rochester