Union-closed Sets Conjecture Holds for Height No More Than 3 and Height No Less Than N-1

TL;DR

This paper proves that the union-closed sets conjecture holds for families with height numbers at most 3 or at least n-1, advancing understanding of the conjecture in specific structural cases.

Contribution

It establishes the validity of the union-closed sets conjecture for families with extreme height numbers, providing a new framework for approaching the conjecture.

Findings

Conjecture holds if height number ≤ 3.

Conjecture holds if height number ≥ n-1.

Provides a framework for proving the conjecture for all height values.

Abstract

For each given union-closed family F of n elements and m sets, we discuss the union-closed sets conjecture from height number of the UC family, which is a natural parameter from lattice theory. In this paper, we call it height number of F(n, m), recorded as H(F). we prove that for any given union-closed family F, union-closed sets conjecture holds if its height number H(F) no more than 3 or no less than n-1. Since the height number H(F) is a positive integer which is bounded between 1 to n. As an attempted approach and framework, if we can prove union-closed sets conjecture holds for all possible value of H(F), then union-closed sets conjecture is true.