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New Lower Bounds for Seven Classical Ramsey Numbers R(3,q)

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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New Lower Bounds for Seven Classical Ramsey Numbers R(3,q)

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  1. 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

  2. 1、Known results on the Ramsey numbers R(3,q) in S.P.Radziszowski, Small Ramsey numbers, Elec.J.Comb.,DS1#10, (2006),1-48 . R(3,25)>=143 R(3,26)>=150 R(3,28)>=164 R(3,29)>=174

  3. 2、The new lower bounds: Theorem: R(3,24)>=143,R(3,25)>=153, R(3,26)>=159,R(3,27)>=167, R(3,28)>=172,R(3,29)>=182,R(3,30)>=187

  4. 3、Three formulas in S.P. Radziszowski,Small Ramsey numbers, Elec.J.Comb.,DS1#10,(2006),1-48 . • R(3,4k+1)>=6R(3,k+1)-5(1) • R(5,k)>=4R(3,k-1)-3 (2) • R(3,k,l+1)>=4R(k,l)-3 (3)

  5. As a consequence of Theorem and the formulas(1),(2)and(3),we obtain Corollary1.R(3,93)>=853, R(3,97)>=913, R(3,101)>=949, R(3,105)>=997, R(3,109)>=1027, R(3,113)>=1087, R(3,117)>=1117 Corollary2.R(5,25)>=569, R(5,26)>=609, R(5,27)>=633, R(5,28)>=665, R(5,29)>=685, R(5,30)>=725, R(5,31)>=745 Corollary3.R(3,3,25)>=569, R(3,3,26)>=609, R(3,3,27)>=633, R(3,3,28)>=665, R(3,3,29)>=685, R(3,3,30)>=725, R(3,3,31)>=745

  6. 4、The algorithm

  7. Given n and the set S1, the algorithm gives the clique number [A2] of Gn[A2] and the first clique of length [Ai]. The detail is listed in the following table

  8. 5、Comments • We also point out other lower bounds such as 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)>=136. These results were obtained by Wang Qingxian and Wang Gongben in 1994 and recorded in S.P.Radziszowski,Small Ramsey numbers, Elec.J.Comb.,DS1#10,(2004),1-48 . According to the reference in it, their paper [WWY1] has not been published. Hence our work verifies these results.

  9. 谢 谢！

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