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arXiv · 2606.29385

Chain Covers in the Boolean Lattice

Zoltán Lóránt Nagy·Balázs Patkós
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Abstract

For integers $1\le r\le n+1$, let $N(n,r)$ denote the least number of chains in the Boolean lattice $B_n=2^{[n]}$ that cover every strict $r$-term chain. The case $r=1$ is the classical chain-decomposition problem and is generalizing Dilworth's theorem and Sperner's theorem. We study two complementary regimes. First, when $r>1$ is fixed and $n\to\infty$. Let $$M(n,r):= \max_{\substack{ a_0+\cdots+a_r=n a_0,a_r\ge 0,\ a_i\ge 1\ (1\le i\le r-1) }} \binom{n}{a_0,\ldots,a_r}.$$ We prove that lower and upper bounds which differ only by a logarithmic factor: \[ M(n,r)\le N(n,r)\le \left(\frac r2+o(1)\right)\log n\cdot M(n,r). \] Second, we consider the near-maximal regime $N(n,n-t)$, where $t>0$ is fixed. We prove that \[ N(n,n-t)= (\gamma_{t-1}+o(1))n!, \] where $\gamma_d$ is the limit of the density of a minimum size subset of the hypercube $Q_n$ that meets all $d$-dimensional subcube.

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Zoltán Lóránt Nagy, Balázs Patkós. 2026-06-28. Chain Covers in the Boolean Lattice. https://arxiv.org/abs/2606.29385

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