Ramsey numbers in complete balanced multipartite graphs. Part I: Set numbers

AbstractThe notion of a graph theoretic Ramsey number is generalised by assuming that both the original graph whose edges are arbitrarily bi-coloured and the sought after monochromatic subgraphs are complete, balanced, multipartite graphs, instead of complete graphs as in the classical definition. We previously confined our attention to diagonal multipartite Ramsey numbers. In this paper the definition of a multipartite Ramsey number is broadened still further, by incorporating off-diagonal numbers, fixing the number of vertices per partite set in the larger graph and then seeking the minimum number of such partite sets that would ensure the occurrence of certain specified monochromatic multipartite subgraphs

Multipartite Graph-Tree Ramsey Numbers

https://digitalcommons.memphis.edu/speccoll-faudreerj/1215/thumbnail.jp

Multipartite Graph-Sparse Graph Ramsey Numbers

https://digitalcommons.memphis.edu/speccoll-faudreerj/1222/thumbnail.jp

On Size Multipartite Ramsey Numbers for Stars Versus Paths and Cycles

Let $K_{l\times t}$ be a complete, balanced, multipartite graph consisting of $l$ partite sets and $t$ vertices in each partite set. For given two graphs $G_1$ and $G_2$, and integer $j\geq 2$, the size multipartite Ramsey number $m_j(G_1,G_2)$ is the smallest integer $t$ such that every factorization of the graph $K_{j\times t}:=F_1\oplus F_2$ satisfies the following condition: either $F_1$ contains $G_1$ or $F_2$ contains $G_2$. In 2007, Syafrizal et al. determined the size multipartite Ramsey numbers of paths $P_n$ versus stars, for $n=2,3$ only. Furthermore, Surahmat et al. (2014) gave the size tripartite Ramsey numbers of paths $P_n$ versus stars, for $n=3,4,5,6$. In this paper, we investigate the size tripartite Ramsey numbers of paths $P_n$ versus stars, with all $n\geq 2$. Our results complete the previous results given by Syafrizal et al. and Surahmat et al. We also determine the size bipartite Ramsey numbers $m_2(K_{1,m},C_n)$ of stars versus cycles, for $n\geq 3,m\geq 2$

An Extension of Ramsey\u27s Theorem to Multipartite Graphs

Ramsey Theorem, in the most simple form, states that if we are given a positive integer l, there exists a minimal integer r(l), called the Ramsey number, such any partition of the edges of K_r(l) into two sets, i.e. a 2-coloring, yields a copy of K_l contained entirely in one of the partitioned sets, i.e. a monochromatic copy of Kl. We prove an extension of Ramsey\u27s Theorem, in the more general form, by replacing complete graphs by multipartite graphs in both senses, as the partitioned set and as the desired monochromatic graph. More formally, given integers l and k, there exists an integer p(m) such that any 2-coloring of the edges of the complete multipartite graph K_p(m);r(k) yields a monochromatic copy of K_m;k . The tools that are used to prove this result are the Szemeredi Regularity Lemma and the Blow Up Lemma. A full proof of the Regularity Lemma is given. The Blow-Up Lemma is merely stated, but other graph embedding results are given. It is also shown that certain embedding conditions on classes of graphs, namely (f , ?) -embeddability, provides a method to bound the order of the multipartite Ramsey numbers on the graphs. This provides a method to prove that a large class of graphs, including trees, graphs of bounded degree, and planar graphs, has a linear bound, in terms of the number of vertices, on the multipartite Ramsey number

The Set Multipartite Ramsey Numbers M_j(P_n, mK_2)

For given two any graph H and G, the set multipartite Ramsey number M_j(G, H) is the smallest integer t such that for every factorization of graph K_(t×j):= F1 F2 so that F1 contains G as a subgraph or F2 contains H as a subgraph. In this paper, we determine  M_j(P_n, mK_2) with j=3,4,5 and m>=2 where P_n denotes a path for n=2,3 vertices and mK_2 denotes a matching (stripes) of size m and pairwise disjoint edges