Description
Abstract:
An antichain (or Sperner family) in the Boolean lattice $B_n$ is a family $\mathcal{A}$ of subsets of $[n]=\{1,2,\dots,n\}$ such that no set in $\mathcal{A}$ is a subset of another.
According to Sperner’s Theorem, the maximum size of an antichain in the Boolean lattice $B_n$ is ${n}\choose{\lfloor n/2 \rfloor}$. We will present a complete answer to the following question:
Which values between 1 and ${n}\choose{\lfloor n/2 \rfloor}$ are attainable as cardinalities of maximal antichains?
(A \emph{maximal} antichain is an antichain $\mathcal{A}$ to which no set from $B_n\setminus\mathcal{A}$ can be added without destroying the antichain property.) This is joint work with Jerry Griggs, Thomas Kalinowski, Michael Schmitz, and Ian Roberts.
Our approach to the above question uses some results on the shadow spectrum $\sigma(k,m)$ which is the set of all $s$ such that there is a $k$-uniform family of $m$ sets whose shadow is of size $s$ and on maximal flat antichains of minimum size. These will also be discussed.
Zoom link: https://zoom.us/j/98309652873?pwd=TkdlNXNRdkZYcHlnRnN3OHhUWkZmQT09