The journey of the union-closed sets conjecture

We survey the state of the union-closed sets conjecture.Comment: Some errors fixed and update

Union-Closed vs Upward-Closed Families of Finite Sets

A finite family \mathrsfs{F} of subsets of a finite set $X$ is union-closed whenever f,g\in\mathrsfs{F} implies f\cup g\in\mathrsfs{F}. These families are well known because of Frankl's conjecture. In this paper we developed further the connection between union-closed families and upward-closed families started in Reimer (2003) using rising operators. With these techniques we are able to obtain tight lower bounds to the average of the length of the elements of \mathrsfs{F} and to prove that the number of joint-irreducible elements of \mathrsfs{F} can not exceed $2{n\choose \lfloor n/2\rfloor}+{n\choose \lfloor n/2\rfloor+1}$ where $|X| = n$.Comment: 40 page