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