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The de Bruijn-Erdos theorem from a Hausdorff measure point of view

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    0511319 - MÚ 2020 RIV HU eng J - Článek v odborném periodiku
    Doležal, Martin - Mitsis, T. - Pelekis, Christos
    The de Bruijn-Erdos theorem from a Hausdorff measure point of view.
    Acta Mathematica Hungarica. Roč. 159, č. 2 (2019), s. 400-413. ISSN 0236-5294
    Grant CEP: GA ČR(CZ) GJ18-01472Y; GA ČR(CZ) GA17-27844S
    Institucionální podpora: RVO:67985840
    Klíčová slova: de Bruijn–Erdős theorem * Hausdorff measure * devil’s staircase * piecewise monotone function
    Kód oboru RIV: BA - Obecná matematika
    Obor OECD: Pure mathematics
    Impakt faktor: 0.538, rok: 2018
    http://dx.doi.org/10.1007/s10474-019-00992-9

    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= { (x, f1(x) , … , fn - 2(x) , α) : x∈ [0 , 1] } , where α is a fixed real number in [0,1] and f1, … , fn - 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 fi, i= 1 , … , 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.
    Trvalý link: http://hdl.handle.net/11104/0301615
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