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)2k-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)2k-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)2k-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)2k-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)2k-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)2k-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)2k-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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