Minimal weight in union-closed families

TL;DR

This paper investigates the minimal total weight of union-closed set families, improving existing bounds under certain conditions and providing constructions that show the bounds are nearly tight.

Contribution

The authors improve Reimer's lower bound on the weight of union-closed families when the family separates points, and they construct examples demonstrating the near tightness of these bounds.

Findings

Improved lower bound on weight when family separates points.

Constructed examples show bounds are tight except in a specific region.

Derived a lower bound on the average degree in point-separating families.

Abstract

Let Omega be a finite set and let S be a set system on Omega. For x in Omega, we denote by d_{S}(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then d(x) \geq |S|/2 for some x in Omega. We consider a related question. Define the weight of S to be w(S)= \sum_{A in S} |A|. Suppose S is union-closed. How small can w(S) be? Reimer showed that w(S) \geq |S| \log_{2} |S| /2, and that this inequality is sharp. In this paper we show how his bound may be improved if we have some additional information about the domain Omega of S: if S separates the points of Omega, then w(S) \geq \binom{|\Omega|}{2}. This is stronger than Reimer's Theorem when Omega > \sqrt{|S|\log_2 |S|}. In addition we construct a family of examples showing the combined bound on w(S) is tight except in the region |\Omega|=\Theta (\sqrt{|S|\log_2 |S|}), where it may be off by a multiplicative factor of 2. Our proof also gives a lower bound on the average degree: if S is a point-separating union-closed family, then the average degree over its domain is at least 1/2 \sqrt{|S| \log_2 |S|}+ O(1), and this is best possible except for a multiplicative factor of 2.