Some results on the multipartite Ramsey numbers m j (C 3, C m , n 1 K 2, n 2 K 2,…, n i K 2)

Yaser Rowshan; Mostafa Gholami; Stanford Shateyi

doi:10.1016/j.heliyon.2022.e11431

Some results on the multipartite Ramsey numbers m _j (C ₃, C _m , n ₁ K ₂, n ₂ K ₂,…, n _i K ₂)

Heliyon. 2022 Nov 3;8(11):e11431. doi: 10.1016/j.heliyon.2022.e11431. eCollection 2022 Nov.

Authors

Yaser Rowshan¹, Mostafa Gholami¹, Stanford Shateyi²

Affiliations

¹ Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 66731-45137, Iran.
² Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa.

Abstract

The graph $K_{j \times t}$ is a graph which is complete and multipartite which includes j partite sets and t vertices in each partite set. The multipartite Ramsey number (M-R-number) $m_{j} (G_{1}, G_{2}, \dots, G_{n})$ is the smallest integer t for the mentioned graphs $G_{1}, G_{2}, \dots, G_{n}$ , in a way which for each n-edge-coloring $(G^{1}, G^{2}, \dots, G^{n})$ of the edges of $K_{j \times t}$ , $G^{i}$ contains a monochromatic copy of $G_{i}$ for at least one i. The size of M-R-number $m_{j} (n K_{2}, C_{7})$ for $j \geq 2$ , $n \leq 6$ , the M-R-number $m_{j} (n K_{2}, C_{7})$ for $j = 2, 3, 4$ , $n \geq 2$ , the M-R-number $m_{j} (n K_{2}, C_{7})$ for each $j \geq 5$ , $n \geq 2$ , the M-R-number $m_{j} (C_{3}, C_{3}, n_{1} K_{2}, n_{2} K_{2}, \dots, n_{i} K_{2})$ for $j \leq 6$ , and $i, n_{i} \geq 1$ , and the size of M-R-number $m_{j} (C_{3}, C_{3}, n K_{2})$ for $j \geq 2$ and $n \geq 1$ have been calculated in various articles hitherto. We acquire some bounds of M-R-number $m_{j} (C_{3}, C_{3}, n_{1} K_{2}, n_{2} K_{2}, \dots, n_{i} K_{2})$ in this essay in which $i, j \geq 2$ , and $n_{i} \geq 1$ , also the size of M-R-number $m_{4} (C_{3}, C_{4}, n K_{2})$ for each $n \geq 1$ is computed in this paper.

Keywords: Cycle; Multipartite Ramsey numbers; Ramsey numbers; Stripes.