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On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice

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arxiv 2411.03400 v2 pith:TBPZQ7BV submitted 2024-11-05 math.CO

Matthew Jenssen , Alexandru Malekshahian , Jinyoung Park This is my paper
classification math.CO
keywords antichainsdedekindnumberproblemsizebetabooleancontained
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abstract

Dedekind's problem, dating back to 1897, asks for the total number $\psi(n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ 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 $B_n$. We obtain detailed estimates for both $\psi(n)$ and the number of antichains of size $\beta \binom{n}{\lfloor n/2 \rfloor}$ for any fixed $\beta>0$. 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 $m$ is contained in a middle layer of $B_n$.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Range of random $\mathbb Z$-homomorphisms on weak expanders

    math.CO 2026-04 unverdicted novelty 7.0 of 10

    Random Z-homomorphisms on weak expanders are O(log log n)-flat with high probability, answering a question of Peled-Samotij-Yehudayoff, and at most 5-valued on Hamming-cube middle layers.

  2. Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo

    math.CO 2026-06 unverdicted novelty 6.0 of 10

    Monte Carlo layer-ratio reconstruction via fixed-layer Markov chains produces the estimate M(10) ≈ 8.936 × 10^78 with uncertainty from cross-n scaling calibrated on known smaller values.

  3. Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo

    math.CO 2026-06 conditional novelty 6.0 of 10

    A layer-ratio Monte Carlo reconstruction estimates M(10) ≈ 8.9360×10^78 and finds a two-shoulder, non-unimodal Whitney-number profile for n=9.

  4. Counting independent sets in percolated graphs via the Ising model

    math.CO 2025-04 unverdicted novelty 6.0 of 10

    An asymptotic expansion is derived for the expected number of independent sets in percolated regular bipartite graphs via the Ising model and cluster expansion, extending prior hypercube work.

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