On k-antichains in the unit n-cube

0524142 - MÚ 2021 RIV HU eng J - Journal Article
Pelekis, Christos - Vlasák, V.
On k-antichains in the unit n-cube.
Publicationes Mathematicae-Debrecen. Roč. 96, 3-4 (2020), s. 503-511. ISSN 0033-3883
R&D Projects: GA ČR(CZ) GJ18-01472Y
Institutional support: RVO:67985840
Keywords : k-antichains * Hausdorff measure * singular function
OECD category: Pure mathematics
Impact factor: 0.636, year: 2020
Method of publishing: Limited access
http://dx.doi.org/10.5486/PMD.2020.8787

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.
Permanent Link: http://hdl.handle.net/11104/0308497