منابع مشابه
Embedding Products of Graphs into Euclidean Spaces
For any collection of graphs G1, . . . , GN we find the minimal dimension d such that the product G1 × · · · ×GN is embeddable into R . In particular, we prove that (K5) and (K3,3) are not embeddable into R, where K5 and K3,3 are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we s...
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We present geometric proofs of Menger’s results on isometrically embedding metric spaces in Euclidean space. In 1928, Karl Menger [6] published the proof of a beautiful characterization of those metric spaces that are isometrically embeddable in the ndimensional Euclidean space E. While a visitor at Harvard University and the Rice Institute in Houston during the 1930-31 academic year, Menger ga...
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M. Hirsch and independently H. Glover have shown that a closed ¿-connected smooth «-manifold M embeds in R2n~> if Mo immerses in A*""*-1, jè2k and 2/gra — 3. Here Mo denotes M minus the interior of a smooth disk. In this note we prove the converse and show also that the isotopy classes of embeddings of M in i?a"-»' are in one-one correspondence with the regular homotopy classes of immersions of...
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Let M be an entire graph in the Euclidean (n+1)-space R. Denote by H , R and |A|, respectively, the mean curvature, the scalar curvature and the length of the second fundamental form of M. We prove that if the mean curvature H of M is bounded then infM |R| = 0, improving results of Elbert and Hasanis-Vlachos. We also prove that if the Ricci curvature of M is negative then infM |A| = 0. The latt...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2020
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2019.105146