Circle grids and bipartite graphs of distances

Abstract

Fort fixed,n+t pointsA 1,A 2,...,A n andB 1,B 2,...,B t are constructed in the plane withO(√n) distinct distancesd(A i B j ) As a by-product we show that the graph of thek largest distances can contain a complete subgraphK t, n withn=Θ(k 2), which settles a problem of Erdős, Lovász and Vesztergombi.

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References

  1. [1]

    P. Erdős: Some applications of graph theory and combinatorial methods to number theory and geometry,Algebraic methods in Graph Theory, Coll. Math. Soc. J. Bolyai 25, (1981), 137–148.

    Google Scholar 

  2. [2]

    G. Elekes:n points in the plane can determinen 3/2 unit circles,Combinatorica 4 (2–3) (1984) 131.

    Google Scholar 

  3. [3]

    L. A. Székely: Private communication.

  4. [4]

    P. Erdős, L. Lovász, K. Vesztergombi: On graphs of large distancesDiscrete and Computational Geometry,4 (1989), 541–549.

    Google Scholar 

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Research partially supported by the Hungarian National Science Fund (OTKA) # 2117.

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Elekes, G. Circle grids and bipartite graphs of distances. Combinatorica 15, 167–174 (1995). https://doi.org/10.1007/BF01200753

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Mathematics Subject Classification (1991)

  • 52 A 38
  • 52 C 05