A counterexample on a group partitioning problem

Anabanti, Chimere (2017) A counterexample on a group partitioning problem. Technical Report. Birkbeck, University of London, London, UK.

Abstract

The Ramsey number Rn(3) is the smallest positive integer such that colouring the edges of a complete graph on Rn(3) vertices in n colours forces the appearance of a monochromatic triangle. We start with a proof that by partitioning the non-identity elements of a finite group into disjoint union of n symmetric product-free sets, we obtain a lower bound for the Ramsey number Rn(3). Exact values of Rn(3) are known for n � 3. The best known lower bound that R4(3) � 51 was given by Chung. In 2006, Kramer gave over 100 pages proof that R4(3) � 62. He then conjectured that R4(3) = 62. In this paper, we say that the Ramsey number Rn(3) is solvable by group partitioning means if there is a finite group G such that |G| + 1 = Rn(3) and G� can be partitioned as a disjoint union of n symmetric product-free sets. We show that Rn(3) (for n � 3) are solvable by group paritioning means while R4(3) is not. Then conjecture that R3(5) � 257 as well as raise the question of which Ramsey numbers are solvable by group partitioning means?

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