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Nearly Equal Distances in the Plane

  • Paul Erdős (a1), Endre Makai (a1) and János Pach (a2)
Abstract

For any positive integer k and ε > 0, there exist nk,ε, ck, e > 0 with the following property. Given any system of n > nk,ε points in the plane with minimal distance at least 1 and any t1, t2…, tk ≥ 1, the number of those pairs of points whose distance is between ti and for some 1 ≤ ik, is at most (n2/2) (1 − 1/(k+1)+ε). This bound is asymptotically tight.

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[1]Erdős, P. (1946) On sets of distances of n points. Amer. Math. Monthly 53, 248250.
[2]Erdős, P. (1965) On extremal problems for graphs and generalized graphs. Israel J. Math. 2, 183190.
[3]Erdős, P., Makai, E., Pach, J. and Spencer, J. (1991) Gaps in difference sets and the graph of nearly equal distances. In: Gritzmann, P. and Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics, the Victor Klee Festschrift, DIMACS Series 4, AMS-ACM, 265273.
[4]Erdős, P. and Purdy, G. (to appear) Some extremal problems in combinatorial geometry. In: Handbook of Combinatorics, Springer-Verlag.
[5]Lovász, L. (1979) Combinatorial problems and exercises, Akad. Kiadó, Budapest, North Holland, Amsterdam—New York—Oxford.
[6]Moser, W. and Pach, J. (1993) Recent developments in combinatorial geometry. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry, Springer-Verlag, Berlin281302.
[7]Pach, J. and Agarwal, P. K. (to appear) Combinatorial Geometry, J. Wiley, New York.
[8]Szemerédi, E. (1978) Regular partitions of graphs. In: Problemes Combinatoires el Théorie de Graphes, Proc. Colloq. Internat. CNRS, Paris399401.
[9]Turán, P. (1941) Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436452. (Hungarian, German summary.)
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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