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A Turán-Type Theorem for Large-Distance Graphs in Euclidean Spaces, and Related Isodiametric Problems


Given a measurable set \(A\subset \mathbb R^d\) we consider the large-distance graph \(\mathcal {G}_A\), on the ground set A, in which each pair of points from A whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the 2d-dimensional Lebesgue measure of the edge set as well as the d-dimensional Lebesgue measure of the vertex set of a large-distance graph in the d-dimensional Euclidean space that contains no copies of a complete graph on k vertices. The former problem may be seen as a continuous analogue of Turán’s classical graph theorem, and the latter as a “graph-theoretic” analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel’s theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest.

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We thank the anonymous referee for his or her comments.

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Correspondence to Jan Hladký.

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M. Doležal: Research supported by the GAČR Project 17-27844S and RVO: 67985840. J. Hladký: Research supported by GAČR Project 18-01472Y and RVO: 67985840. J. Kolář: Research supported by the EPSRC grant EP/N027531/1 and by RVO: 67985840. C. Pelekis: Research supported by the Czech Science Foundation, Grant Number GJ16-07822Y, by GAČR Project 18-01472Y and RVO: 67985840.

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Doležal, M., Hladký, J., Kolář, J. et al. A Turán-Type Theorem for Large-Distance Graphs in Euclidean Spaces, and Related Isodiametric Problems. Discrete Comput Geom 66, 281–300 (2021).

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  • Turán’s theorem
  • Isodiametric problem
  • Distance graph

Mathematics Subject Classification

  • 05C63
  • 51K99
  • 05C35
  • 51M16