The ith Ramsey number for matchings
Given graphs \(G_ 1,...,G_ k\) we denote by \(r_ i(G_ 1,...,G_ k)\) the least integer p so that, if the complete p-partite graph K(i,...,i) with each vertex set of size i has its edges colored with k colors, \(c_ 1,...,c_ k\), then, for some index j, there is a subgraph of K(i,...,i) isomorphic to \(G_ j\) with all of its edges colored \(c_ j\). One easily checks that the usual Ramsey number associated with \(G_ 1,...,G_ k\) is, in this notation, \(r_ 1(G_ 1,...,G_ k)\). The Ramsey index for \(G_ 1,...,G_ k\) is defined to be the least index i such that \[ r_ i(G_ 1,...,G_ k)=\lim_{j\to \infty}(G_ 1,...,G_ k); \] and is denoted by \(i(G_ 1,...,G_ k)\). The main results of this paper include: Theorem 9. Let \(k,m_ 1,...,m_ k\) be positive integers, then \[ i(m_ 1K_ 2,...,m_ kK_ 2) = \sum^{k}_{j=1}(m_ j-1)+1. \]
scientific article; zbMATH DE number 3970797