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New Lower Bounds for Seven Classical Ramsey Numbers R(3,q)
Kang Wu
South China Normal University
Wenlong Su
Guangxi University Wuzhou Branch
Haipeng Luo , Xiaodong Xu
Guangxi Academy of Sciences
2006 年 8 月
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11 、、 Known results on the Ramsey Known results on the Ramsey numbers R(3,q) in numbers R(3,q) in S.P.Radziszowski, SS.P.Radziszowski, Small Ramsey numbers, Elec.J.Comb.,DS1#10, mall Ramsey numbers, Elec.J.Comb.,DS1#10, (2006),1-48(2006),1-48 . .
R(3,25)>=143R(3,25)>=143
R(3,26)>=150R(3,26)>=150
R(3,28)>=164R(3,28)>=164
R(3,29)>=174R(3,29)>=174
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22 、、 The new lower bounds:The new lower bounds:
Theorem:Theorem:
R(3,24)>=143,R(3,25)>=153,R(3,24)>=143,R(3,25)>=153,
R(3,26)>=159,R(3,27)>=167,R(3,26)>=159,R(3,27)>=167,
R(3,28)>=172,R(3,29)>=182,R(3,30)>=187R(3,28)>=172,R(3,29)>=182,R(3,30)>=187
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33 、、 Three formulas in Three formulas in S.P. RadziszS.P. Radziszowski,Small Ramsey numbers, Elec.J.owski,Small Ramsey numbers, Elec.J.Comb.,DS1#10,(2006),1-48Comb.,DS1#10,(2006),1-48 . .
• R(3,4k+1)>=6R(3,k+1)-5R(3,4k+1)>=6R(3,k+1)-5 (1)(1)
• R(5,k)>=4R(3,k-1)-3 (2)R(5,k)>=4R(3,k-1)-3 (2)
• R(3,k,l+1)>=4R(k,l)-3 (3)R(3,k,l+1)>=4R(k,l)-3 (3)
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As a consequence of Theorem and the As a consequence of Theorem and the formulas(1),(2)and(3),we obtainformulas(1),(2)and(3),we obtain
Corollary1.Corollary1. R(3,93)>=853, R(3,97)>=913,R(3,93)>=853, R(3,97)>=913,
R(3,101)>=949, R(3,105)>=997,R(3,101)>=949, R(3,105)>=997,
R(3,109)>=1027, R(3,113)>=1087, R(3,109)>=1027, R(3,113)>=1087, R(3,117)>=1117R(3,117)>=1117
Corollary2.Corollary2. R(5,25)>=569, R(5,26)>=609,R(5,25)>=569, R(5,26)>=609,
R(5,27)>=633, R(5,28)>=665,R(5,27)>=633, R(5,28)>=665,
R(5,29)>=685, R(5,30)>=725, R(5,31)>=745R(5,29)>=685, R(5,30)>=725, R(5,31)>=745
Corollary3.Corollary3. R(3,3,25)>=569, R(3,3,26)>=609,R(3,3,25)>=569, R(3,3,26)>=609,
R(3,3,27)>=633, R(3,3,28)>=665,R(3,3,27)>=633, R(3,3,28)>=665,
R(3,3,29)>=685, R(3,3,30)>=725, R(3,3,31)>=745R(3,3,29)>=685, R(3,3,30)>=725, R(3,3,31)>=745
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44 、、 The algorithmThe algorithm1 ) G i v e n i n t e r g e r 5n , l e t [ ]2
nm . G i v e n a 2 - p a r t i t i o n 1 2S S S o f
[ 1 , ]S m w h e r e b o t h 1S a n d 2S a r e n o n e m p t y . L e t i iq S f o r 1 , 2i . L e t 1i .
2 ) S e t { , }i n i iA x x Z x S o r n x S . S o r t iA l e x i c o g r a p h i c a l l y . A s s u m e
t h a t 1 2{ , , . . . , }iA x x . S e t [ ] 1 , 1iA j . 3 ) F o r j ix S , f i n d ( ) { , }i j i j j id x y y A y x a n d x y A . I f ( ) 0i jd x g o t o 5 ) .
4 ) F i n d t h e iA – c o l o r e d c h a i n s t a r t i n g w i t h j ix S . I f ( ) [ ]i j il x A , l e t [ ] ( ) 1i i jA l x a n d p r i n t o u t t h i s c h a i n .
5 ) I n c r e a s e j b y 1 . I f ij q , g o t o 3 ) .
6 ) L e t [ ] 1i ik A . I n c r e a s e i b y 1 . I f 2i , g o t o 2 ) .
7 ) P r i n t o u t 1 2( 1 , 1 ) 1R k k n a n d t h e a l g o r i t h m i s t e r m i n a t e d .
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Given n and the set SGiven n and the set S11, the algorithm , the algorithm gives the clique number [Agives the clique number [A22] of G] of Gnn[A[A22] ] and the first clique of length [Aand the first clique of length [Aii]. The ]. The detail is listed in the following tabledetail is listed in the following table
n Elements of S1 [A2] The first clique [A2] in Gn[A2]
91 1,4,10,15,23,28,
37,42
15 2,5,7,13,16,18,21,24,27,29,32,38,
40,43,45
97 1,5,13,22,29,33,
41,47
16 2,4,8,10,12,18,20,28,36,38,46,63,
73,81,89,91
105 1,4,6,21,23,38,
40,49,51
17 2,5,7,10,12,15,17,20,22,27,29,32,
34,37,39,44,46
108 1,4,6,15,20,29,
32,42,45,54
18 2,5,7,10,12,21,23,33,35,46,51,58,
60,69,72,74,82,85
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121 1,3,9,17,25,27,
32,40,46,51
19 2,4,6,8,12,16,20,26,30,49,61,65,
69,73,80,84,99,103,117
124 1,3,5,12,19,26,
37,46,48,54,62
20 2,4,6,8,10,17,24,31,42,51,53,59,
67,75,81,83,92,103,110,117
135 1,3,5,9,16,23,33,
37,44,52,64
21 2,4,6,8,10,14,21,28,38,42,49,57,
69,76,88,96,103,107,117,124,131
142 1,3,5,7,15,28,34,
36,45,47,58,71
22 2,4,6,8,10,12,14,22,35,41,43,52,54,
65,78,91,102,104,113,115,121,134
152 1,3,5,9,13,21,28,
32,39,46,63,70
23 2,4,6,8,10,14,18,20,26,37,44,51,61,
68,75,87,94,98,105,125,129,136,148
158 1,3,5,13,22,24,28,
32,39,46,63,70
24 2,4,6,8,10,18,25,27,29,33,37,44,52,56,
58,60,67,101,103,110,117,124,143,150
166 1,3,5,7,16,18,29,31,
39,53,66,72,80
25 2,4,6,8,10,12,14,23,25,36,38,44,46,50,58,
71,93,101,107,120,128,134,142,153,155
171 1,3,5,7,9,20,31,42,
53,64,66,77,79
26 2,4,6,8,10,12,14,16,18,29,40,51,62,73,
75,86,88,90,101,103,114,116,127,138,
149,160
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181 1,3,5,7,9,13,25,33,
44,62,73,79,81,85
27 2,4,6,8,10,12,14,16,18,29,40,51,62,73,75,
86,88,90,101,103,114,116,127,138,149,160
186 1,3,5,7,9,11,21,36,
44,46,59,61,74,78,93
28 2,4,6,8,10,12,14,16,18,20,22,32,47,55,57,70,
72,85,89,104,117,123,136,138,151,153,161,176
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55 、、 CommentsComments• We also point out other lower bounds such aWe also point out other lower bounds such a
s R(3,17)>=92,R(3,18)>=98,R(3,19)>=106,R(3,2s R(3,17)>=92,R(3,18)>=98,R(3,19)>=106,R(3,20)>=109,R(3,21)>=122,R(3,22)>=125,R(3,23)>=0)>=109,R(3,21)>=122,R(3,22)>=125,R(3,23)>=136. These results were obtained by Wang Qi136. These results were obtained by Wang Qingxian and Wang Gongben in 1994 and recorngxian and Wang Gongben in 1994 and recorded in ded in S.P.Radziszowski,Small Ramsey numbers, ElS.P.Radziszowski,Small Ramsey numbers, Elec.J.Comb.,DS1#10,(2004),1-48ec.J.Comb.,DS1#10,(2004),1-48 . According to the . According to the reference in it, their paper [WWY1] has not bereference in it, their paper [WWY1] has not been published. Hence our work verifies these ren published. Hence our work verifies these results. esults.
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谢 谢!谢 谢!