RAMSEY GUEST LISTS

DIAGONAL RAMSEY NUMBERS
"WATERMAN CONJECTURE of 2010"

As of September 2010...I changed the lettering assigned in my formulas to better comply with standard lettering....however, all my conjectured Ramsey values remained totally unchanged.

"Erdo"s asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens."

Perhaps we could just use my formula below, instead ?

The classic Ramsey problem can be phrased in terms of guests at a party. What is the minimum number of guests at a party that need to be invited so that either at least three guests will all know each other or be mutual strangers? Answer 6. Suppose we want not a threesome but a foursome who either know each other or are mutual strangers? Erdos and Ramsey theorists have proved that 18 guests are required. For a fivesome, that answer is believed to be 43 to 49. For a sixsome...102 to 165.
I propose the following values (for numbers up to a trillion) are for additional levels, without any proof..."the Waterman diagonal Ramsey conjecture". Values have been obtained through the use of my following formula

R = s(s-1) 2 ^k-s
where R = Ramsey numberk = discussion group size and s = size of the unit group

s = 3

k	Ramsey number = R	known limits	s	(k-1)	2 ^k-s
3	6	6	3	2	1
4	18	18	3	3	2
5	48	43-49	3	4	4
6	120	102-165	3	5	8
7	288	205-540	3	6	16
8	672	282-1870	3	7	32
9	1,536	565-6588	3	8	64
10	3,456	798-23556	3	9	128
11	7,680	1597-184755	3	10	256
12	16,896	1837-705431	3	11	512
13	36,864	2557-2704155	3	12	1,024
14	79,872	2989-10400599	3	13	2,048
15	172,032	5485-40116599	3	14	4,096
16	368,640	5605-155117519	3	15	8,192
17	786,432	8917-601080389	3	16	16,384
18	1,671,168	11005-2333606219	3	17	32,768
19	3,538,944	17885-9075135299	3	18	65,536
20	7,471,104		3	19	131,072
21	15,728,640		3	20	262,144
22	33,030,144		3	21	524,288
23	69,208,016		3	22	1,048,576
24	144,703,488		3	23	2,097,152
25	301,989,888		3	24	4,194,304
26	629,145,600		3	25	8,388,608
27	1,308,622,848		3	26	16,777,216
28	2,717,908,992		3	27	33,554,432
29	5,637,144,576		3	28	87,108,864
30	11,676,942,336		3	29	134,217,728
31	24,159,191,040		3	30	268,435,456
32	49,928,994,816		3	31	536,870,912
33	103,079,215,104		3	32	1,073,741,824
34	212,600,881,152		3	33	2,147,483,648
35	438,086,664,192		3	34	4,294,967,296
36	901,943,132,160		3	35	8,589,934,592

The classic Ramsey problem only relates 3 guests as a basic unit. The stipulation is that any 3 guests which arrive together must be either strangers or be acquainted. However, what if this unit group of 3 guests were extended to a unit group of 4 guests. That is, invited guests arrive in blocks of 4, all of which must be either mutually acquainted or all strangers to one another. Later, they can mingle indisciminately. The charts below indicate these increased unit groups for all additional Ramsey values up to a trillion.
I also have conject regarding a more generalized formula for these "supplimentary" Ramsey numbers... conjectured...

R = s, (s-1), (s-2) 2 ^k-s , (s-3) 2 ^k-s , etc. where s = unit group size
k = discussion group size

s = 4

k	R	s	(s-1)	(s-2) 2 ^k-s	(s-3) 2 ^k-s
4	24	4	3	2	1
5	128	4	4	4	2
6	640	4	5	8	4
7	3,072	4	6	16	8
8	14,336	4	7	32	16
9	65,536	4	8	64	32
10	294,912	4	9	128	64
11	1,310,720	4	10	256	128
12	5,767,168	4	11	512	256
13	25,165,824	4	12	1024	512
14	109,051,904	4	13	2,048	1,024
15	469,762,048	4	14	4,096	2,048
16	2,013,265,920	4	15	8,192	4,096
17	8,589,934,592	4	16	16,384	8,192
18	36,507,222,016	4	17	32,768	16,384
19	154,618,822,656	4	18	65,536	32,768
20	652,835,028,992	4	19	131,072	65,536

s = 5

k	R	s	(s-1)	(s-2) 2 ^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-a
5	120	5	4	3	2	1
6	1,200	5	5	6	4	2
7	11,520	5	6	12	8	4
8	107,520	5	7	24	16	8
9	983,040	5	8	48	32	16
10	8,847,360	5	9	96	64	32
11	78,643,200	5	10	192	128	64
12	692,060,160	5	11	384	256	128
13	6,039,797,760	5	12	768	512	256
14	52,344,913,920	5	13	1,536	1,024	512
15	450,971,566,080	5	14	3,072	2,048	1,024

s = 6

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s
6	720	6	5	4	3	2	1
7	13,824	6	6	8	6	4	2
8	258,048	6	7	16	12	8	4
9	4,718,592	6	8	32	24	16	8
10	84,934,656	6	9	64	48	32	16
11	1,509,949,440	6	10	128	96	48	32
12	26,575,110,144	6	11	256	192	96	64
13	463,856,467,968	6	12	512	384	192	128

s = 7

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s	(s-6) 2 ^k-s
7	5,040	7	6	5	4	3	2	1
8	188,160	7	7	10	8	6	4	2
9	6,881,220	7	8	20	16	12	8	4
10	247,726,080	7	9	40	32	24	16	8
11	8,808,038,400	7	10	80	64	48	32	16
12	310,042,951,680	7	11	160	128	96	64	32

s = 8

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s	(s-6) 2 ^k-s	(s-7) 2 ^k-s
8	40,320	8	7	6	5	4	3	2	1
9	2,949,120	8	8	12	10	8	6	4	2
9	212,336,640	8	9	24	20	16	12	8	4
10	15,099,494,400	8	10	48	40	32	24	16	8

s = 9

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s	(s-6) 2 ^k-s	(s-7) 2 ^k-s	(s-8) 2 ^k-s
9	362,880	9	8	7	6	5	4	3	2	1
10	52,254,720	9	9	14	12	10	8	6	4	2
11	7,431,782,400	9	10	28	24	20	16	12	8	4

s = 10

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s	(s-6) 2 ^k-s	(s-7) 2 ^k-s	(s-8) 2 ^k-s	(s-9) 2 ^k-s
10	3,628,800	10	9	8	7	6	5	4	3	2	1
11	1,032,192,000	10	10	16	14	12	10	8	6	4	2
12	290,665,267,200	10	11	32	28	24	20	16	12	8	4

s = 11

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s	(s-6) 2 ^k-s	(s-7) 2 ^k-s	(s-8) 2 ^k-s	(s-9) 2 ^k-s	(s-10) 2 ^k-s
11	39,916,800	11	10	9	8	7	6	5	4	3	2	1
12	22,481,141,760	11	11	18	16	14	12	10	8	6	4	2

s = 12

k	R	s	(s-1)	(s-2) 2^k-s	(s-3) 2 ^k-s	(s-4) 2 ^k-s	(s-5) 2 ^k-s	(s-6) 2 ^k-s	(s-7) 2 ^k-s	(s-8) 2 ^k-s	(s-9) 2 ^k-s	(s-10) 2 ^k-s	(s-11) 2 ^k-s
12	479,001,600	12	11	10	9	8	7	6	5	4	3	2	1
13	535,088,332,800	11	12	20	18	16	14	12	10	8	6	4	2

The remaining Ramsey numbers less than a trillion can also be seen as R! (factorial)

k	R
13	6,227,020,800
14	87,178,291,200

In total then, there are 34 basic Ramsey numbers
(unit group is 3) less than a trillion.
There are an additional 58 Ramsey numbers
(unit groups > 3) less than a trillion.

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