# Dimension-Free Bounds for the Union-Closed Sets Conjecture

## Abstract

**:**

## 1. Introduction

**Conjecture 1**

## 2. Main Results

**Theorem 1.**

**Proposition 1.**

**Corollary 1.**

## 3. Proof of Theorem 1

**Lemma 1**

- (13) follows since $\frac{a+b}{c+d}\ge min\{\frac{a}{c},\frac{b}{d}\}$ for $a,b\ge 0,c,d>0$, and $H\left({X}_{i}\right|{X}^{i-1})=0$ implies ${X}_{i}$ is a deterministic function of ${X}^{i-1}$ and, hence, ${g}_{i}({P}_{{X}^{i-1}},\rho )=0$;

## 4. Proof of Proposition 1

## 5. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Yu, L.
Dimension-Free Bounds for the Union-Closed Sets Conjecture. *Entropy* **2023**, *25*, 767.
https://doi.org/10.3390/e25050767

**AMA Style**

Yu L.
Dimension-Free Bounds for the Union-Closed Sets Conjecture. *Entropy*. 2023; 25(5):767.
https://doi.org/10.3390/e25050767

**Chicago/Turabian Style**

Yu, Lei.
2023. "Dimension-Free Bounds for the Union-Closed Sets Conjecture" *Entropy* 25, no. 5: 767.
https://doi.org/10.3390/e25050767