The saturation spectrum for antichains of subsets

TL;DR

This paper characterizes the sizes of maximal antichains in Boolean lattices and extends the Kruskal-Katona theorem to determine possible shadow sizes of families of k-sets, revealing new combinatorial bounds.

Contribution

It provides a complete characterization of maximal antichain sizes and introduces an extended Kruskal-Katona theorem for shadow sizes of k-set families.

Findings

Characterization of integers m for maximal antichains in B_n.

Complete answer for shadow sizes when t ≤ k+1.

Asymptotic bound on the largest non-shadow size, approximately √2 k^{3/2}.

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 $\sqrt 2k^{3/2}+\sqrt[4]{8}k^{5/4}+O(k)$.