On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice
Abstract
Abstract Dedekind's problem, dating back to 1897, asks for the total number of antichains contained in the Boolean lattice on elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics, and as a result, obtain several new results on the number and typical structure of antichains in . We obtain detailed estimates for both and the number of antichains of size for any fixed . We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size is contained in a middle layer of .
Fields of Study & Subjects
Advanced Algebra and Logic
Advanced Topology and Set Theory
Rough Sets and Fuzzy Logic
Publication details
- Journal / source
- Journal of the London Mathematical Society
- Volume
- 114
- Issue
- 1
- Publication date
- 1 July 2026
- Language
- EN