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On a problem of P. Erdős. (English) Zbl 0612.05032
For $$\alpha$$ real, $$0<\alpha <2$$, and n an integer, let G(n,$$\alpha)$$ be the graph whose vertices are the points of the unit sphere $$S^{n- 1}=\{x=(x_ 1,...,x_ n)\in {\mathbb{R}}^ n:\sum (x_ i)^ 2=1\}$$, and which has an edge between two points if and only if they are a distance of $$\alpha$$ apart. $$\chi$$ (G(n,$$\alpha)$$) is the chromatic number of the graph G(n,$$\alpha)$$, finite because $$S^{n-1}$$ is a compact metric space. It is shown that for each $$\alpha$$, $$\chi$$ (G(n,$$\alpha)$$)$$\to \infty$$ as $$n\to \infty$$. The proof uses known results on the chromatic number of Kneser graphs in a critical manner.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C99 Graph theory
##### Keywords:
chromatic number; Kneser graphs
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##### References:
 [1] B. BOLLOBÁS: Extremal Graph Theory. Academic Press, London-New York-San Francisco, 1978. · Zbl 0844.05054
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