Size Multipartite Ramsey Numbers for K4-e Versus all Graphs up to 4 Vertices

Abstract

Let G and H be two simple subgraphs of s j K ×. The smallest positive integer s such that any red and blue colouring of s j K × has a copy of red G or a blue H is called the multipartite Ramsey number of G and H. It is denoted by) , (H G m j. This paper presents exact values for) , (2 G B m j where G is a isolate vertex free graph up to four vertices.

$mj(P_4,G)$ for all Graphs G on 4 Vertices

“…that H R contains no red P 4 and H B contains no blue K 1 , 3 . As H B contains no blue K 1 , 3 both v 1,1 , v 1,2 will satisfy deg R (v 1,1 ) ≥ 4 and deg R (v 1,2 ) ≥ 4.…”

“…But as m 3 (P 3 ,C 3 ) = 2 we get that there exists red P 3 with end points x and y. Let z and w be two points not in this red P 3 and not belonging to the partite sets x, y belong to. But then as H R contains no red P 4 , we will obtain that x, z, y, w, x is a blue C 4 , a contradiction.…”

“…Given any two coloring (consisting of say red and blue colors) of the edges of a graph K j×s , we say that K j×s → (P 4 ,G), if there exists a red copy of P 4 in K j×s or a blue copy of Gin K j×s . The size Ramsey multipartite number m j (P 4 ,G) is defined as the smallest natural number t such that K j×t → (P 4 ,G) (see [1,3,4,5,6,7] for general cases of m j (H,G)). In this paper, we exhaustively find m j (P 4 ,G) for all 11 non-isomorphic graphs G on 4 vertices.…”

On a Ramsey Problem Involving Quadrilaterals

“…In order to show, m 5 As there is no blue C 4 , (v 2,2 ,v 3,1 ) and (v 4,2 ,v 2,1 ) have to be red edges. Next as there is no red C 4 , (v 2,2 ,v 4,1 ) has to be a blue edge.…”

“…In the last four decades most of the Ramsey numbers R(H,K) have been studied in detail for |V (H)| <7 and |V (K)| <7 (see [6]). The size Ramsey multipartite number m j (H,K) is defined as the smallest natural number t such that K j×t → (H,K) (see [1,3,5,7]). In this paper we concentrate on determining multipartite Ramsey numbers m j (C 4 ,G) for all possible proper subgraphs G of K 4 .…”