Lubell mass and induced partially ordered sets

TL;DR

This paper proves a bound on the size of subset families avoiding induced copies of a given poset, confirming a conjecture and advancing understanding of extremal set theory related to partially ordered sets.

Contribution

It establishes a universal bound for families avoiding induced posets, confirming a conjecture and linking poset structure with extremal combinatorial properties.

Findings

Existence of a universal constant c(P) for each poset P.

Families avoiding induced P have bounded weighted size.

Confirms a conjecture of Lu and Milans.

Abstract

We prove that for every partially ordered set $P$, there exists $c(P)$ such that every family $\mathcal{F}$ of subsets of $[n]$ ordered by inclusion and which contains no induced copy of $P$ satisfies $\sum_{F\in \mathcal{F}}1/{n\choose |F|}\leq c(P)$. This confirms a conjecture of Lu and Milans.