科研之友
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摘要
An r-edge coloring of a graph or hypergraph G = (V, E) is a map c : E {0,, r 1}. Extending results of Rado and answering questions of Rado, Gyarfas and Sarkozy we prove that the vertex set of every r-edge colored countably infinite complete k-uniform hypergraph can be partitioned into r monochromatic tight paths with distinct colors (a tight path in a k-uniform hypergraph is a sequence of distinct vertices such that every set of k consecutive vertices forms an edge); for all natural numbers r and k there is a natural number M such that the vertex set of every r-edge colored countably infinite complete graph can be partitioned into M monochromatic kth powers of paths apart from a finite set (a kth power of a path is a sequence v(0), v(1),... of distinct vertices such that 1 <= vertical bar i-j vertical bar <= k implies that viva is an edge); the vertex set of every 2-edge colored countably infinite complete graph can be partitioned into 4 monochromatic squares of paths, but not necessarily into 3; the vertex set of every 2-edge colored complete graph on col can be partitioned into 2 monochromatic paths with distinct colors.
- 出版日期2017-8
