An upper bound for union-closed family size

TL;DR

This paper establishes an upper bound on the size of union-closed families of sets, characterizes when equality holds, and provides related bounds involving binomial coefficients and parameters for combinatorial set families.

Contribution

It introduces a new upper bound for union-closed families and characterizes the extremal families achieving equality, extending previous combinatorial bounds.

Findings

Proved that the size of a union-closed family is at most the sum of binomial coefficients up to its length.

Characterized the families that achieve the maximum size.

Derived bounds involving parameters p and k for related combinatorial sums.

Abstract

Let $\mathcal{A}$ be a union-closed family of sets with universe $\bigcup_{A \in \mathcal{A}}A = [n] = \{1,\cdots,n\}$ and length $\ell$. We prove that $|\mathcal{A}| \leq \sum_{i=0}^{\ell} \binom{n}{i}$, with equality if and only if $\mathcal{A} = \bigcup_{i=0}^{\ell}\binom{[n]}{n-i}$. Additionally, by showing that $|\mathcal{A}| \leq \frac{\ell^p-1}{\ell-1}+2^n(1-2^{-\ell})^p$ for any nonnegative integer $p$, we establish for all integers $1 \leq k \leq n$ that $\sum_{i=0}^k \binom{n}{i} \leq \frac{k^{\hat{p}}-1}{k-1}+2^n(1-2^{-k})^{\hat{p}}$, where $\hat{p}=\lfloor (n-k)/\log_2(\frac{k}{1-2^{-k}})\rfloor + 1$.