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Embedding products of graphs into Euclidean spaces

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Skopenkov M.

Fundamenta Mathematicae

2003

tom 179

nr 3

191-198

For any collection of graphs $G₁,...,G_N$ we find the minimal dimension d such that the product $G₁ × ... × G_N$ is embeddable into $ℝ^d$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3,3})ⁿ$ are not embeddable into $ℝ^{2n}$, where K₅ and $K_{3,3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $Lk O → S^{2n-1}$, where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.

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1 Fundamenta Mathematicae

1 Skopenkov M.

1 2003

Wyniki wyszukiwania

Embedding products of graphs into Euclidean spaces