Monochromatic coverings and tree Ramsey numbers
The main result of this paper is the following Ramsey type theorem. Let \(k=3\) or 4 and let \(n\) be a natural number not divisible by \(k-1\). Consider any edge \(k\)-coloring of the complete graph \(K_ p\) where \(p=(k-1) (n-1)+2\). Then \(K_ p\) admits \(k-1\) monochromatic connected subgraphs \(G_ 1, \dots, G_{k-1}\) so that \[ | V (G_ 1) | \geq n+1 \text{ and } V(G_ 1) \cup V(G_ 2) \cup \cdots \cup V (G_{k- 1})=V (K_ p). \] Some partial results are also obtained for \(k>4\).
scientific article; zbMATH DE number 562682