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<oai_dc:dc xmlns:dc='http://purl.org/dc/elements/1.1/' xmlns:oai_dc='http://www.openarchives.org/OAI/2.0/oai_dc/' xmlns:xsi='http://www.w3.org/2001/XMLSchema-instance' xsi:schemaLocation='http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd'>
<dc:creator>Anabanti, C.S.</dc:creator>
<dc:date>2023-12-29</dc:date>
<dc:description xml:lang='en'><p>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) &gt; 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) &gt; 257.</p></dc:description>
<dc:identifier>10.56415/qrs.v31.12</dc:identifier>
<dc:source>Quasigroups and Related Systems  (2) 165-174</dc:source>
<dc:subject>Ramsey numbers</dc:subject>
<dc:subject>product-free sets</dc:subject>
<dc:subject>groups</dc:subject>
<dc:subject>Partition</dc:subject>
<dc:title>The Ramsey number R4(3) is not solvable by group partition means</dc:title>
<dc:type>info:eu-repo/semantics/article</dc:type>
</oai_dc:dc>