# Families in posets minimizing the number of comparable pairs

## TL;DR

This paper investigates the centeredness property in posets, disproves a conjecture for certain cases, and extends results to the poset of subspaces of a finite field, contributing to extremal combinatorics.

## Contribution

It disproves the conjecture that {0,1,...,k}^n has the centeredness property for all k, and extends the property to the poset of subspaces of {F}_q^n.

## Key findings

- Disproved the conjecture for all k.
- Identified the range of M where the property holds.
- Extended the centeredness property to subspace posets.

## 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 $M$, among 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 $\{0,1,\ldots,k\}^n$ 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 $\mathbb{F}_q^n$ has the centeredness property. Several open questions are also given.