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

Abstract

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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References

  1. 1.

    Bollobás, B.: Measure graphs. J. Lond. Math. Soc. 21(3), 401–412 (1980)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bollobás, B.: Extremal Graph Theory. Dover, Mineola (2004)

    Google Scholar 

  3. 3.

    Bonnesen, T., Fenchel, W.: Theory of Convex Bodies. BCS Associates, Moscow (1987)

    Google Scholar 

  4. 4.

    Burago, Yu.D., Zalgaller, V.A.: Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften, vol. 285. Springer, Berlin (1988)

    Google Scholar 

  5. 5.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics. CRC Press, Boca Raton (2015)

    Google Scholar 

  6. 6.

    Doležal, M., Hladký, J., Kolář, J., Mitsis, T., Pelekis, C., Vlasák, V.: A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems. Acta Math. Univ. Comenian. 88(3), 625–629 (2019)

    MathSciNet  Google Scholar 

  7. 7.

    Kahle, M., Martinez-Figueroa, F.: The chromatic number of random Borsuk graphs. Random Struct. Algorithms (2019). https://doi.org/10.1002/rsa.20897

  8. 8.

    Katona, G.O.H.: Continuous versions of some extremal hypergraph problems. In: Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), vol. 2. Colloq. Math. Soc. János Bolyai, vol. 18, pp. 653–678. North-Holland, Amsterdam (1978)

  9. 9.

    Katona, G.O.H.: Continuous versions of some extremal hypergraph problems II. Acta Math. Acad. Sci. Hungar. 35(1–2), 67–77 (1980)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Katona, G.O.H.: Turán’s graph theorem, measures and probability theory. In: Pintz, J., et al. (eds.) Number Theory, Analysis, and Combinatorics (Proc. Paul Turán Memorial Conference, Budapest, 2011). De Gruyter Proceedings in Mathematics, pp. 167–176. De Gruyter, Berlin (2014)

    Google Scholar 

  11. 11.

    Kollár, J., Rónyai, L., Szabó, T.: Norm-graphs and bipartite Turán numbers. Combinatorica 16(3), 399–406 (1996)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lovász, L.: Large Networks and Graph Limits. American Mathematical Society Colloquium Publications, vol. 60. American Mathematical Society, Providence (2012)

    Google Scholar 

  13. 13.

    Lovász, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory Ser. B 96(6), 933–957 (2006)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mantel, W.: Problem XXVIII. Wiskundige Opgaven 10, 60–61 (1907). (in Dutch)

    MATH  Google Scholar 

  15. 15.

    Matoušek, J.: Using the Borsuk–Ulam Theorem. Universitext. Springer, Berlin (2003)

    Google Scholar 

  16. 16.

    Pelekis, C.: A generalized isodiametric problem. Geombinatorics 25(4), 151–167 (2016)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Pólya, G.: Sur la symétrisation circulaire. C. R. Acad. Sci. Paris 230, 25–27 (1950)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Shabanov, L.E., Raigorodskii, A.M.: Turán type results for distance graphs. Discrete Comput. Geom. 56(3), 814–832 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Simonovits, M., Sós, V.T.: Ramsey–Turán theory. Discrete Math. 229(1–3), 293–340 (2001)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Turán, P.: On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436–452 (1941). (in Hungarian)

    MathSciNet  Google Scholar 

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Acknowledgements

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 (2020). https://doi.org/10.1007/s00454-020-00183-2

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Keywords

  • Turán’s theorem
  • Isodiametric problem
  • Distance graph

Mathematics Subject Classification

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