Chain-dependent Conditions in Extremal Set Theory

TL;DR

This paper introduces a new method for extremal set theory problems involving chain-dependent conditions, enabling proofs of classic theorems and new results on family sizes and chain counts.

Contribution

It presents a novel weight-based double counting method for chain-dependent conditions, applicable to various extremal set theory problems.

Findings

Proved a new theorem on families avoiding certain subset relations.

Established a method to maximize the number of ll-chains in a family.

Provided a unified approach to classic extremal set theory results.

Abstract

In extremal set theory our usual goal is to find the maximal size of a family of subsets of an $n$-element set satisfying a condition. A condition is called chain-dependent, if it is satisfied for a family if and only if it is satisfied for its intersections with the $n!$ full chains. We introduce a method to handle problems with such conditions, then show how it can be used to prove three classic theorems. Then, a theorem about families containing no two sets such that $A\subset B$ and $\lambda \cdot |A| \le |B|$ is proved. Finally, we investigate problems where instead of the size of the family, the number of $\ell$-chains is maximized. Our method is to define a weight function on the sets (or $\ell$-chains) and use it in a double counting argument involving full chains.