Abstract
ABSTRACTThe Boolean lattice consists of all subsets of partially ordered under the containment relation. Sperner's Theorem states that the largest antichain of the Boolean lattice is given by a middle layer: the collection of all sets of size , or also, if is odd, the collection of all sets of size . Given , choose each subset of with probability independently. We show that for every constant , the largest antichain among these subsets is also given by a middle layer, with probability tending to 1 as tends to infinity. This is best possible, and we also characterize the largest antichains for every constant . Our proof is based on some new variations of Sapozhenko's graph container method.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callSimilar Papers
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
