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

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

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

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)

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

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 .

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

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

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

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.

谢 谢！谢 谢！