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. A lower bound on Rn(3) is obtainable by partitioning the nonidentity elements of a finite group into disjoint union of n symmetric product-free sets. Exact values of Rn(3) are known for n 6 3. The best known lower bound that R4(3) > 51 was given by Chung. In 2006, Kramer gave a proof of over 100 pages that R4(3) 6 62. He then conjectured that R4(3) = 62. We say that the Ramsey number Rn(3) is solvable by group partition means if there is a finite group G such that jGj+1 = Rn(3) and Gnf1g can be partitioned as a union of n symmetric product-free sets. For n 6 3, the Ramsey number Rn(3) is solvable by group partition means. Some authors believe that R4(3) not be solvable by a group partition approach. We prove this here. We also show that any finite group G whose size is divisible by 3 cannot enjoy Gnf1g written as a disjoint union of its symmetric product-free sets. We conclude with a conjecture that R5(3) > 257.