A proof of the union-close set conjecture

TL;DR

This paper proves the union-close set conjecture by introducing new concepts like universe, communities, and spots, demonstrating that in any finite universe, a spot exists with at least half the density in an induced community.

Contribution

The paper provides a formal proof of the union-close set conjecture using novel terminology and framework, advancing understanding in combinatorial set theory.

Findings

Proof of the union-close set conjecture for finite universes

Introduction of the concepts of universe, communities, and spots

Establishment of the existence of a spot with density at least 1/2

Abstract

In this paper, we introduce the notion of the universe, induced communities, and cells with their corresponding spots. Using this language, we formulate and prove the union close set conjecture by showing that for any finite universe $\mathbb{U}$ and any induced community $\mathcal{M}_{\mathbb{U}}$ there exist some spot $a\in \mathbb{U}$ such that the density \begin{align} \mathcal{D}_{\mathcal{M}_{\mathbb{U}}}(a)\geq \frac{1}{2}.\nonumber \end{align}