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(s1) 2 ^{ks} where R = Ramsey number 
k  
3  6  6  3  2  1 
4  18  18  3  3  2 
5  48  4349  3  4  4 
6  120  102165  3  5  8 
7  288  205540  3  6  16 
8  672  2821870  3  7  32 
9  1,536  5656588  3  8  64 
10  3,456  79823556  3  9  128 
11  7,680  1597184755  3  10  256 
12  16,896  1837705431  3  11  512 
13  36,864  25572704155  3  12  1,024 
14  79,872  298910400599  3  13  2,048 
15  172,032  548540116599  3  14  4,096 
16  368,640  5605155117519  3  15  8,192 
17  786,432  8917601080389  3  16  16,384 
18  1,671,168  110052333606219  3  17  32,768 
19  3,538,944  178859075135299  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... 
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 
2 ^{ks}  2 ^{ks} 
2 ^{ka}  
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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks}  
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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks}  
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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 

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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 

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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 

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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 

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 
2^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 
2 ^{ks} 

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