Chain Covers in the Boolean Lattice

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 a general upper bound $ N(n,n-t)\le \frac{n!}{t}$ using the inversion number of the permutations modulo $t$. This is exact for $t=2$, giving $N(n,n-2)=n!/2$, and asymptotically exact for $t=3$, giving $N(n,n-3)=\left(\frac13+o(1)\right)n!.$ The matching lower bound for $t=3$, and stronger lower bounds for all fixed $t$, come from subcube-hitting problems originated from Kostochka and vertex-Turán problems.