The saturation spectrum for antichains of subsets
Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz
TL;DR
The paper characterizes the saturation spectrum of maximal antichains in the Boolean lattice by introducing the shadow spectrum $\sigma(t,k)$ and deriving its exact form for $t\le k+1$, enabling a near-complete description of attainable sizes close to the Sperner maximum. A key contribution is a three-level flat-antichain construction that yields wide intervals of achievable sizes, together with precise overlap arguments that cover large ranges of $m$ for most $n$. The authors also determine the largest shadow-gap $\psi(k)$ asymptotically as $\psi(k)=\sqrt{2}\,k^{3/2}+\sqrt[4]{8}\,k^{5/4}+O(k)$, and they provide detailed proofs for both the upper and lower parts of the spectrum, including computer-assisted checks for small cases. The work connects Sperner theory to Kruskal--Katona shadow theory, advances understanding of saturation phenomena in $B_n$, and leaves several open directions, including general $\sigma(t,k)$ for larger $t$ and related saturated poset problems.
Abstract
Extending a classical theorem of Sperner, we characterize the integers $m$ such that there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers $t$ and $k$, we ask which integers $s$ have the property that there exists a family $\mathcal F$ of $k$-sets with $\lvert\mathcal F\rvert=t$ such that the shadow of $\mathcal F$ has size $s$, where the shadow of $\mathcal F$ is the collection of $(k-1)$-sets that are contained in at least one member of $\mathcal F$. We provide a complete answer for $t\leqslant k+1$. Moreover, we prove that the largest integer which is not the shadow size of any family of $k$-sets is $\sqrt 2k^{3/2}+\sqrt[4]{8}k^{5/4}+O(k)$.