A continuous analogue of Erdős' k-Sperner theorem

0517694 - MÚ 2021 RIV US eng J - Journal Article
Mitsis, T. - Pelekis, Christos - Vlasák, V.
A continuous analogue of Erdős' k-Sperner theorem.
Journal of Mathematical Analysis and Applications. Roč. 484, č. 2 (2020), č. článku 123754. ISSN 0022-247X. E-ISSN 1096-0813
R&D Projects: GA ČR(CZ) GJ18-01472Y
Institutional support: RVO:67985840
Keywords : Chains * k-Sperner families * Hausdorff measure * Lebesgue measure
OECD category: Pure mathematics
Impact factor: 1.583, year: 2020
Method of publishing: Limited access
https://doi.org/10.1016/j.jmaa.2019.123754

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 show that the 1-dimensional Hausdorff measure of a chain in the unit n-cube is at most n, and that the bound is sharp. Given this result, we consider the problem of maximising the n-dimensional Lebesgue measure of a measurable set A⊂[0,1]n subject to the constraint that it satisfies H1(A∩C)≤κ for all chains C⊂[0,1]n, where κ is a fixed real number from the interval (0,n]. We show that the measure of A is not larger than the measure of the following optimal set: Aκ⁎={(x1,…,xn)∈[0,1]n:n−κ2≤∑i=1nxi≤n+κ2}. Our result may be seen as a continuous counterpart to a theorem of Erdős, regarding k-Sperner families of finite sets.
Permanent Link: http://hdl.handle.net/11104/0302997