A Proof of a Conjecture on Bipartite Ramsey Numbers B (2,2,3)

Yaser Rowshan, Mostafa Gholami and Stanford Shateyi
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Yaser Rowshan: Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 66731-45137, Iran
Mostafa Gholami: Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 66731-45137, Iran
Stanford Shateyi: Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa

Mathematics, 2022, vol. 10, issue 5, 1-9

Abstract: The bipartite Ramsey number B ( n 1 , n 2 , … , n t ) is the least positive integer b , such that any coloring of the edges of K b , b with t colors will result in a monochromatic copy of K n i , n i in the i -th color, for some i , 1 ≤ i ≤ t . The values B ( 2 , 5 ) = 17 , B ( 2 , 2 , 2 , 2 ) = 19 and B ( 2 , 2 , 2 ) = 11 have been computed in several previously published papers. In this paper, we obtain the exact values of the bipartite Ramsey number B ( 2 , 2 , 3 ) . In particular, we prove the conjecture on B ( 2 , 2 , 3 ) which was proposed in 2015—in fact, we prove that B ( 2 , 2 , 3 ) = 17 .

Keywords: Ramsey numbers; bipartite Ramsey numbers; Zarankiewicz number (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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