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 ----- Any reply is welcome!