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Let S be a metric space and let k be a positive integer. Define χ ̂(k)(S) to be the smallest positive integer m such that for every k×m array D=(D ij ) of positive real numbers, S can be colored with the colors 1,2,…,m such that no two points of distance D ij are both colored j. We improve the best upper bound known on χ ̂(k)(R) from 32 k k! to ⌈4ek⌉, where e is the base...
A mixed hypergraph consists of two families of subsets: the edges and the co-edges. In a coloring every co-edge has at least two vertices of the same color, and every edge has at least two vertices of different colors. The largest and smallest possible number of colors in a coloring is termed the upper and lower chromatic numbers, respectively. In this paper we investigate co-hypergraphs i.e., the...
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