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On k-antichains in the unit n-cube
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SYSNO ASEP 0524142 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title On k-antichains in the unit n-cube Author(s) Pelekis, Christos (MU-W) SAI, RID
Vlasák, V. (CZ)Source Title Publicationes Mathematicae-Debrecen. - : Kossuth Lajos Tudomanyegyetem - ISSN 0033-3883
Roč. 96, 3-4 (2020), s. 503-511Number of pages 9 s. Language eng - English Country HU - Hungary Keywords k-antichains ; Hausdorff measure ; singular function Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects GJ18-01472Y GA ČR - Czech Science Foundation (CSF) Method of publishing Limited access Institutional support MU-W - RVO:67985840 UT WOS 000530645200015 EID SCOPUS 85091172898 DOI 10.5486/PMD.2020.8787 Annotation A chain in the unit n-cube is a set C ⊂ [0, 1]n such that for every x = (x1, . . . , xn) and y = (y1, . . . , yn) in C, we either have xi ≤ yi for all i ∈ [n], or xi ≥ yi for all i ∈ [n]. We consider subsets A, of the unit n-cube [0, 1]n, that satisfy card(A ∩ C) ≤ k, for all chains C ⊂ [0, 1]n, where k is a fixed positive integer. We refer to such a set A as a k-antichain. We show that the (n − 1)-dimensional Hausdorff measure of a k-antichain in [0, 1]n is at most kn and that the bound is asymptotically sharp. Moreover, we conjecture that there exist k-antichains in [0, 1]n whose (n − 1)-dimensional Hausdorff measure equals kn, and we verify the validity of this conjecture when n = 2. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2021 Electronic address http://dx.doi.org/10.5486/PMD.2020.8787
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