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arXiv · 2106.02226

The saturation spectrum for antichains of subsets

Jerrold R. Griggs·Thomas Kalinowski·Uwe Leck·Ian T. Roberts·Michael Schmitz
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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 $\sqrt 2k^{3/2}+\sqrt[4]{8}k^{5/4}+O(k)$.

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Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz. 2021-06-04. The saturation spectrum for antichains of subsets. https://doi.org/10.1007/s11083-022-09622-6

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