Variations on the theme of repeated distances


We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ℝd, d≥4. We also give asymptotically tight upper bounds on the total number of occurrences of the “favourite” distances fromn points in ℝd, d≥4. Related results are proved for distances determined byn disjoint compact convex sets in ℝ2.

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  1. [1]

    D. Avis, P. Erdős, andJ. Pach: Repeated distances in space,Graphs and Combinatorics,4 (1988), 207–217.

    Google Scholar 

  2. [2]

    B. Bollobás:Extremal Graph Theory, Academic Press, London,1978.

    Google Scholar 

  3. [3]

    B. Bollobás, P. Erdős, S. Simonovits, andE. Szemerédi: Extremal graphs without large forbidden subgraphs, in:Advances in Graph Theory, Ann. Discr. Math.,3 (1978), 29–41.

    Google Scholar 

  4. [4]

    W. G.Brown, and F.Harary: Extremal digraphs, in:Combinatorial Theory and its Applications (Erdős et al, eds.),Coll. Math. Soc. J. Bolyai,4, North-Holland, 1970, 135–198.

  5. [5]

    W. G. Brown, andM. Simonovits: Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures,Discr. Math.,48 (1984), 147–162.

    Google Scholar 

  6. [6]

    K. L.Clarkson, H.Edelsbrunner, L. J.Guibas, M.Sharir, and E.Welzl: Combinatorial complexity bounds for arrangements of curves and surfaces, to appear inDiscr. Comput. Geom.

  7. [7]

    H. Edelsbrunner:Algorithms in Combinatorial Geometry: Springer-Verlag, Heidelberg,1987.

    Google Scholar 

  8. [8]

    P. Erdős: On sets of distances ofn points,Amer. Math. Monthly,53 (1946), 248–250.

    Google Scholar 

  9. [9]

    P. Erdős: On sets of distances ofn points in Euclidean space,Publ. Math. Inst. Hung. Acad. Sci.,5 (1960), 165–169.

    Google Scholar 

  10. [10]

    P. Erdős: On some applications of graph theory to geometry,Canadian J. Math.,19 (1967), 968–971.

    Google Scholar 

  11. [11]

    P. Erdős, D. Hickerson, andJ. Pach: A problem of Leo Moser about repeated distances on the sphere,Amer. Math. Monthly. 96 (1989), 569–575.

    Google Scholar 

  12. [12]

    P. Erdős, andA. H. Stone: On the structure of linear graphs,Bull. Amer. Math. Soc.,52 (1946), 1087–1091.

    Google Scholar 

  13. [13]

    K. Kedem, R. Livne, J. Pach, andM. Sharir: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles,Discr. Comput. Geom.,1 (1986), 59–71.

    Google Scholar 

  14. [14]

    T. Kővári, V. T. Sós, andP. Turán: On a problem of Zarankiewicz,Colloquium Math.,3 (1954), 50–57.

    Google Scholar 

  15. [15]

    W. O. J. Moser, andJ. Pach:100 Research Problems in Discrete Geomentry, McGill University, Montreal,1987.

    Google Scholar 

  16. [16]

    J. Spencer, E. Szemerédi, andW. T. Trotter, Jr.: Unit distances in Euclidean plane, in:Graph Theory and Combinatorics, Academic Press, London,1984, 293–303.

    Google Scholar 

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At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.

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Erdős, P., Pach, J. Variations on the theme of repeated distances. Combinatorica 10, 261–269 (1990).

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AMS subject classification (1980)

  • 52 A 37
  • 52 A 40