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On the number of edges in induced subgraphs of a special distance graph

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Abstract

We obtain new estimates for the number of edges in induced subgraphs of a special distance graph.

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References

  1. A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters,” in Discrete Geometry and Algebraic Combinatorics, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2014), Vol. 625, pp. 93–109.

    Google Scholar 

  2. A. M. Raigorodskii, “Coloring distance graphs and graphs of diameters,” in Thirty Essays on Geometric Graph Theory (Springer, New York, 2013), pp. 429–460.

    Chapter  Google Scholar 

  3. A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56 1, 107–146 (2001) [RussianMath. Surveys 56 1, 103–139 (2001)].

    Article  MathSciNet  MATH  Google Scholar 

  4. A. M. Raigorodskii, “On the chromatic numbers of spheres in Euclidean spaces,” Dokl. Ross. Akad. Nauk 432 2, 174–177 (2010) [Dokl. Math. 81 3, 379–382 (2010)].

    MathSciNet  MATH  Google Scholar 

  5. A. M. Raigorodskii, “On the chromatic numbers of spheres in Rn,” Combinatorica 32 1, 111–123 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Balogh, A. Kostochka, and A. Raigorodskii, “Coloring some finite sets in Rn,” Discuss. Math. Graph Theory 33 1, 25–31 (2013).

  7. L. I. Bogolyubskii, A S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of random subgraphs in some sequences of graphs,” Dokl. Ross. Akad. Nauk 457 4, 383–387 (2014) [Dokl. Math. 90 1, 462–465 (2014)].

    MathSciNet  MATH  Google Scholar 

  8. L. I. Bogolyubskii, A S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, “Independence numbers and chromatic numbers of the random subgraphs of some distance graphs,” Mat. Sb. 206 10, 3–36 (2015) [Sb. Math. 206 10, 1340–1374 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Pach and P. K. Agarwal, Combinatorial Geometry (JohnWiley and Sons, New York, 1995).

    Book  MATH  Google Scholar 

  10. L. A. Székely, “Erdős on unit distances and the Szemerédi–Trotter theorems,” in Paul Erdős and HisMathematics, II, Bolyai Soc. Math. Stud. (János BolyaiMath. Soc., Budapest, 2002), Vol. 11, pp. 649–666.

    Google Scholar 

  11. A. Soifer, The Mathematical Coloring Book. Mathematics of Coloring and the Colorful Life of its Creators (Springer, New York, 2009).

    MATH  Google Scholar 

  12. V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, in The DolcianiMath. Exp. (Math. Assoc. America, Washington, DC, 1991), Vol. 11.

    Google Scholar 

  13. V. Boltyanski, H. Martini, and P. S. Soltan, Excursions into Combinatorial Geometry, in Universitext (Springer-Verlag, Berlin, 1997).

    Book  MATH  Google Scholar 

  14. A. M. Raigorodskii, “Three lectures on the Borsuk partition problem,” in Surveys in Contemporary Mathematics, LondonMath. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 2008), Vol. 347, pp. 202–248.

    Google Scholar 

  15. A. M. Raigorodskii, “Around Borsuk’s hypothesis,” Sovrem. Mat. Fundam. Napravl. 23, 147–164 (2007) [J. Math. Sci. (New York) 154 4, 604–623 (2008)].

    MATH  Google Scholar 

  16. R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey Theory (JohnWiley and Sons, New York, 1990).

    MATH  Google Scholar 

  17. Z. Nagy, “A certain constructive estimate of the Ramsey number,” Mat. Lapok 23, 301–302 (1972).

    MathSciNet  Google Scholar 

  18. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam–New York–Oxford, 1977; Radio i Svyaz’, Moscow, 1979), Parts I, II.

    MATH  Google Scholar 

  19. L. Bassalygo, G. Cohen, and G. Zémor, “Codes with forbidden distances,” DiscreteMath. 213, 3–11 (2000).

    MathSciNet  MATH  Google Scholar 

  20. E. E. Demekhin, A. M. Raigorodskii, and O. I. Rubanov, “Distance graphs having large chromatic numbers and containing no cliques or cycles of a given size,” Mat. Sb. 204 4, 49–78 (2013) [Sb. Math. 204 4, 508–538 (2013)].

    Article  MathSciNet  MATH  Google Scholar 

  21. K. A. Mikhailov and A. M. Raigorodskii, “On the Ramsey numbers for complete distance graphs with vertices in 0, 1n,” Mat. Sb. 200 12, 63–80 (2009) [Sb. Math. 200 12, 1789–1806 (2009)].

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to F. A. Pushnyakov.

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Original Russian Text © F. A. Pushnyakov, 2016, published in Matematicheskie Zametki, 2016, Vol. 99, No. 4, pp. 550–558.

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Pushnyakov, F.A. On the number of edges in induced subgraphs of a special distance graph. Math Notes 99, 545–551 (2016). https://doi.org/10.1134/S0001434616030275

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  • DOI: https://doi.org/10.1134/S0001434616030275

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