Let $T=\{1,2,\dots,2n+1\}$. What is the largest $k$ such that we can choose $k$ subsets of size $n$ and $k$ subsets of size $n+1$ of $T$ so that no chosen subset contains another?


$k=\binom{2n}{n-1}$ is possible: Choose all sets of size $n$ containing $1$, and all sets of size $n+1$ not containing $1$. Does it follow from Sperner's theorem that this is the best possible?



Non-containing subsets of two sizes
Let $\mathcal{A}$ be any collection of subsets of $\{1,\dots,m\}$ such that no subset in $\mathcal{A}$ is contained in another. Let $a\_i$ be the number of $i$-element subsets in $\mathcal{A}$. A complete characterization of the sequences $(a\_0,a\_1,\dots)$ appears as Theorem 2.2 in <https://pdfs.semanticscholar.org/da0b/1a3430e4be45119921b8b8a6578b355bb2f6.pdf>. In principle it should be possible to use this result to answer your question, but I did not work out the details.


