Diagonal Ramsey numbers in multipartite graphs related to stars

Multipartite Ramsey Numbers

For a graph G, a partiteness k ≥ 2 and a number of colours c, we define the multipartite Ramsey number r k (G) as the minimum value m such that, given any colouring using c colours of the edges of the complete balanced k-partite graph with m vertices in each partite set, there must exist a monochromatic copy of G. We show that the question of the existence of r k (G) is tied up with what monoch...

Off-diagonal hypergraph Ramsey numbers

The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-subsets of {1, . . . , N} contains a red set of size s or a blue set of size n, where a set is red (blue) if all of its k-subsets are red (blue). A k-uniform tight path of size s, denoted by Ps, is a set of s vertices v1 < · · · < vs in Z, and all s−k+1 edges of the form {vj , vj+1, . . . , vj+k−1}. Let rk(Ps...

Ramsey numbers in complete balanced multipartite graphs. Part II: Size numbers

The 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...

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

The 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...

Size Ramsey Numbers of Stars versus 4-Chromati Graphs