On combinatorial properties of spheres in euclidean spaces

Abstract

For λ>√2 there exists a triangle-free graphG such that for nod canG be imbedded into thed-dimensional unit sphere with two points joined if and only if their distance is >λ.

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References

  1. [1]

    B. Alspach, M. Rosenfeld, On embedding triangle-free graphs in unit spheres,Discrete Math.,19 (1977), 103–111.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    P. Erdős, F. Galvin, A. Hajnal, On set-systems having large chromatic number and not containing prescribed subsystems,Infinite and Finite Sets (A. Hajnal, R. Rado and V. T. Sós eds.) Budapest, Hungary 1975, 425–514.

    Google Scholar 

  3. [3]

    D. C. Larman, A Triangle Free Graph Which Cannot be √3-imbedded in any Euclidean Unit Sphere,Journal of Combinatorial Theory, (A) 24 (1978), 162–169.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    F. P. Ramsey, On a Problem of Formal Logic,Proc. of the London Math. Soc. (2) 30 (1930), 264–286.

    Article  Google Scholar 

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Rödl, V. On combinatorial properties of spheres in euclidean spaces. Combinatorica 4, 345–349 (1984). https://doi.org/10.1007/BF02579146

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AMS subject classification (1980)

  • 05 C 99
  • 51 N 20