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diagonal Ramsey number R(n,n)= 4k+2, R(5,5)=46

  • To: mathgroup at smc.vnet.net
  • Subject: [mg109865] diagonal Ramsey number R(n,n)= 4k+2, R(5,5)=46
  • From: a boy <a.spring.boy at gmail.com>
  • Date: Thu, 20 May 2010 06:39:02 -0400 (EDT)

In my opinion, the red-blue critical graph (R(n,n)-1 nodes) for
diagonal Ramsey number R(n,n)=r have two self-symmetry below:
1. It exists at least one node  that  the number of its red edges is
same as blue, equal to (r-2)/2, so r-2 is even;
2.  In the critical graph, the number of all red edges is as many as
blue, each equal to (r-2)(r-1)/4, so r-2=4k or r-1=4k.
In a word, diagonal Ramsey number R(n,n) = 4k+2.
R(2,2)=2, R(3,3)=6, R(4,4)=18 follow this form all. It has been known
that R(5,5) is between [43, 49], only 46 has the form 4k+2, so I think
R(5,5)=46

References: http://mathworld.wolfram.com/RamseyNumber.html
       http://en.wikipedia.org/wiki/Ramsey's_theorem

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Any reply is welcome!


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