The de Bruijn–Erdős theorem from a Hausdorff measure point of view


Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erdős, we consider curves in the unit n-cube \([0,1]^n\) of the form

$$A= \bigl\{(x,f_1(x),\ldots,f_{n-2}(x),\alpha): x\in [0,1] \bigr\},$$

where \(\alpha\) is a fixed real number in [0,1] and \(f_1, \ldots, f_{n-2}\) are injective measurable functions from [0,1] to [0,1]. We refer to such a curve A as an n-de Bruijn–Erdős-set. Under the additional assumption that all functions \(f_i, i=1,\ldots,n-2,\) are piecewise monotone, we show that the Hausdorff dimension of A is at most 1 as well as that its 1-dimensional Hausdorff measure is at most n-1. Moreover, via a walk along devil’s staircases, we construct a piecewise monotone n-de Bruijn–Erdős-set whose 1-dimensional Hausdorff measure equals n-1.

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  1. 1.

    Anderson, I.: Combinatorics of Finite Sets, Corrected reprint of the 1989 edition, Dover Publications, Inc. Mineola, NY (2002)

    Google Scholar 

  2. 2.

    C.J. Bishop and Y. Peres, Fractals in Probability and Analysis, Cambridge Studies in Advanced Mathematics, 162. Cambridge University Press (Cambridge, 2017)

  3. 3.

    A. Blokhuis, A.E. Brouwer, A. Chowdhury, P. Frankl, T. Mussche, B. Patkós and T. Szőnyi, A Hilton–Milner theorem for vector spaces, Electron. J. Combin., 17 (2010), Research Paper 71, 12 pp

  4. 4.

    Bogachev, V.I.: Measure Theory, vol. 1. Springer (2007)

  5. 5.

    Cameron, P.J.: Combinatorics: Topics. Algorithms, Cambridge University Press, Techniques (1994)

    Google Scholar 

  6. 6.

    F. Coen, N. Gillman, T. Keleti, D. King and J. Zhu, Large sets with small injective projections, arXiv:1906.06288 (2019)

  7. 7.

    N. G. de Bruijn and Erdős, On a combinatorial problem, Indag. Math., 10 (1948) 421–423

  8. 8.

    Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24, 1–37 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    V. Eiderman and M. Larsen, A rare plane set with Hausdorff dimension 2, arXiv:1904.09034 (2019)

  10. 10.

    Engel, K.: A continuous version of a Sperner-type theorem. Elektron. Informationsverarb. Kybernet. 22, 45–50 (1986)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    K. Engel, Sperner Theory, Encyclopedia of Mathematics and its Applications, 65, Cambridge University Press (Cambridge, 1997)

  12. 12.

    K. Engel, T. Mitsis and C. Pelekis, A fractal perspective on optimal antichains and intersecting subsets of the unit \(n\)-cube, arXiv:1707.04856 (2017)

  13. 13.

    K. Engel, T. Mitsis, C. Pelekis and C. Reiher, Projection inequalities for antichains, Israel J. Math. (to appear), arXiv:1812.06496

  14. 14.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press (1992)

  15. 15.

    K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd. (Chichester, 1990)

  16. 16.

    J. Foran, The length of the graph of a one to one function from \([0,1]\) to \([0,1]\), Real Anal. Exchange, 25 (1999/00), 809–816

  17. 17.

    P. Frankl and N. Tokushige, The Katona theorem for vector spaces, J. Comb. Theory, Ser. A, 120 (2013), 1578–1589

    MathSciNet  Article  Google Scholar 

  18. 18.

    P. Frankl and R. M. Wilson, The Erdős–Ko–Rado theorem for vector spaces, J. Combin. Theory, Ser. A, 43 (1986), 228–236

    MathSciNet  Article  Google Scholar 

  19. 19.

    G. O. H. Katona, Continuous versions of some extremal hypergraph problems, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 653–678, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978

  20. 20.

    G. O. H. Katona, Continuous versions of some extremal hypergraph problems. II, Acta Math. Acad. Sci. Hungar., 35 (1980), 67–77

    MathSciNet  Article  Google Scholar 

  21. 21.

    A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag (New York, 1995)

  22. 22.

    Klain, D.A., Rota, G.C.: A continuous analogue of Sperner's theorem. Comm. Pure Appl. Math. 50, 205–223 (1997)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Katchalski, M., Meshulam, R.: An extremal problem for families of pairs of subspaces. European J. Combin. 15, 253–257 (1994)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Zaanen, A.C., Luxemburg, W.A.J.: A real function with unusual properties, [Solution to Problem 5029]. Amer. Math. Monthly 70, 674–675 (1963)

    MathSciNet  Article  Google Scholar 

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We are grateful to the anonymous referee for bringing to our attention references [6] and [9].

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Corresponding author

Correspondence to Ch. Pelekis.

Additional information

Research of Doležal was supported by the GAČR project 17-27844S and RVO: 67985840.

Research of Pelekis was supported by the GAČR project 18-01472Y and RVO: 67985840.

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Doležal, M., Mitsis, T. & Pelekis, C. The de Bruijn–Erdős theorem from a Hausdorff measure point of view. Acta Math. Hungar. 159, 400–413 (2019).

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Key words and phrases

  • de Bruijn–Erdős theorem
  • Hausdorff measure
  • devil’s staircase
  • piecewise monotone function

Mathematics Subject Classification

  • 05D05
  • 28A78
  • 26A30