The size, multipartite Ramsey numbers for C3 versus all graphs up to 4 vertices

Abstract

Let the star on n vertices, namely K1,n−1 be denoted by Sn. If every two coloring of the edges of a complete balanced multipartite graph Kj×s there is a copy of Sn in the first color or a copy of Sm in the second color, then we will say Kj×s → (Sn, Sm). The size Ramsey multipartite number mj(Sn, Sm) is the smallest natural number s such that Kj×s → (Sn, Sm). In this paper, we obtain the exact values of the size Ramsey numbers mj(Sn, Sm) for n, m 3 and j 3.

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

“…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.…”

“…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. The summary of our findings is illustrated in Table 1.…”

On a Ramsey Problem Involving Quadrilaterals

“…Therefore, we will get m 4 (C 4 ,B 2 ) ≥ 3. To show that m 3 (C 4 ,B 2 ) ≤ 3 consider any coloring consisting of (red , blue) given by K 3×4 = H R ⊕H B , such that H R contains no red C 4 Also in the remaining 6 vertices (in S c ∩ T c ) must contain a blue P 3 as m 3 (C 4 ,P 3 ) = 2. Thus, without loss of generality we may assume that v 1,3 v 2,3 v 3,3 is a blue P 3 .…”

“…Next, applying m 3 (C 4 ,C 3 ) = 3 to K 3×3 consisting of the first three elements of the three partite sets, we obtain a blue B 2 containing v 1,4 . Hence the claim follows.…”

“…It is worth noting that all values of 4 4 ( , ) j m C P and m j (C 4 ,C 3 ) (see [4])are currently known. Also all values of m j (C 4 ,K 1,3 + x) are known as m j (C 4 ,C 3 ) = m j (C 4 ,K 1,3 + x) for any integer j.…”