Sarka Petrickova (Illinois Math) Families in posets minimizing the number of comparable pairs Abstract: Given a poset $P$, we say a family $\mathcal{F} \subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$, amongst all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ (where $(A_1,\dots,A_n)\le (B_1,\dots,B_n)$ if $A_i\le B_i$ for each $i\in [n]$) also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbf{F}_q^n$ has the centeredness property. This is joint work with Jozsef Balogh and Adam Zsolt Wagner. 
