Generalized Ramsey numbers for paths in 2-chromatic graphs
Summary: Chung and Liu have defined the d-chromatic Ramsey number as follows. Let \(1\leq d\leq c\) and let \(t=\left( \begin{matrix} c\\ d\end{matrix} \right)\). Let 1,2,...,t be the ordered subsets of d colors chosen from c distinct colors. Let \(G_ 1,G_ 2,...,G_ t\) be graphs. The d-chromatic Ramsey number denoted by \(r^ c_ d(G_ 1,G_ 2,...,G_ t)\) is defined as the least number p such that, if the edges of the complete graph \(K_ p\) are colored in any fashion with c colors, then for some i, the subgraph whose edges are colored in the ith subset of colors contains a \(G_ i\). In this paper it is shown that \(r^ 3_ 2(P_ i,P_ j,P_ k)=[(4k+2j+i-2)/6]\) where \(i\leq j\leq k<r(P_ i,P_ j)\), \(r^ 3_ 2\) stands for a generalized Ramsey number on a 2-colored graph and \(P_ i\) is a path of order i.
scientific article; zbMATH DE number 3959476