A continuous analogue of Erdős' k-Sperner theorem

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00517694" target="_blank" >RIV/67985840:_____/20:00517694 - isvavai.cz</a>
Alternative codes found
RIV/00216208:11320/20:10422171
Result on the web
<a href="https://doi.org/10.1016/j.jmaa.2019.123754" target="_blank" >https://doi.org/10.1016/j.jmaa.2019.123754</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jmaa.2019.123754" target="_blank" >10.1016/j.jmaa.2019.123754</a>

Result language
angličtina
Original language name
A continuous analogue of Erdős' k-Sperner theorem
Original language description
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.
Czech name
—
Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GJ18-01472Y" target="_blank" >GJ18-01472Y: Graph limits and inhomogeneous random graphs</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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