mathematics Article A Proof of a Conjecture on Bipartite Ramsey Numbers B(2, 2, 3) Yaser Rowshan 1 , Mostafa Gholami 1 and Stanford Shateyi 2, *   Citation: Rowshan, Y.; Gholami, M.; Shateyi, S. A Proof of a Conjecture on Bipartite Ramsey Numbers B(2, 2, 3). Mathematics 2022, 10, 701. https:// doi.org/10.3390/math10050701 Academic Editors: Janez Žerovnik and Darja Rupnik Poklukar Received: 7 December 2021 Accepted: 23 December 2021 Published: 23 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 66731-45137, Iran; y.rowshan@iasbs.ac.ir (Y.R.); gholami.m@iasbs.ac.ir (M.G.) 2 Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa * Correspondence: stanford.shateyi@univen.ac.za 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 1. Introduction 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 existence of such a positive integer is guaranteed by a result of Erd˝ os and Rado [1]. The Zarankiewicz number z(K m,n , t) is defined as the maximum number of edges in any subgraph G of the complete bipartite graph K m,n , such that G does not contain K t,t as a subgraph. Zarankiewicz numbers and related extremal graphs have been studied by many authors, including Kóvari [2], Reiman [3], and Goddard, Henning, and Oellermann in [4]. The study of bipartite Ramsey numbers was initiated by Beineke and Schwenk in 1976 [5], and continued by others, in particular Exoo [6], Hattingh, and Henning [7]. The following exact values have been established: B(2, 5)= 17 [8], B(2, 2, 2, 2)= 19 [9], B(2, 2, 2)= 11 [6]. In the smallest open case for five colors, it is known that 26 ≤ B(2, 2, 2, 2, 2) ≤ 28 [9]. One can refer to [2,9–14] and it references for further studies. Collins et al. in [8] showed that 17 ≤ B(2, 2, 3) ≤ 18, and in the same source made the following conjecture: Conjecture 1 ([8]). B(2, 2, 3)= 17. We intend to get the exact value of the multicolor bipartite Ramsey numbers B(2, 2, 3). We prove the following result: Theorem 1. B(2, 2, 3)= 17. In this paper, we are only concerned with undirected, simple, and finite graphs. We follow [15] for terminology and notations not defined here. Let G be a graph with vertex set V(G) and edge set E(G). The degree of a vertex v ∈ V(G) is denoted by deg G (v), or simply by deg(v). The neighborhood N G (v) of a vertex v is the set of all vertices of G adjacent to v and satisfies | N G (v)| = deg G (v). The minimum and maximum degrees of vertices of G are denoted by δ(G) and Δ(G), respectively. Additionally, the complete bipartite graph with bipartition ( X, Y), where | X| = m and |Y| = n, is denoted by K m,n . We use [ X, Y] to denote the set of edges between the bipartition ( X, Y) of G. Let G =( X, Y) be a bipartite graph Mathematics 2022, 10, 701. https://doi.org/10.3390/math10050701 https://www.mdpi.com/journal/mathematics