[Paper Review] Minimum density of union-closed families

Igor Balla|arXiv (Cornell University)|Jun 2, 2011

Limits and Structures in Graph Theory4 references3 citations

TL;DR

This paper proves Wójcik's conjecture that the minimum density of a union-closed family with largest set of size $n$ is asymptotically $\log_2 n / (2n)$ as $n \to \infty$. Using Reimer's inequality on average set size in union-closed families, the authors establish a lower bound of $\log_2 n / (2n)$, which matches Wójcik's conjectured asymptotic value, and derive a corollary showing that for $n \geq 16$, some element appears in at least $\sqrt{(\log_2 n)/n} \cdot |\mathscr{F}|/2$ sets.

ABSTRACT

Let F be a finite union-closed family of sets whose largest set contains n elements. In \cite{Wojcik92}, Wojcik defined the density of F to be the ratio of the average set size of F to n and conjectured that the minimum density over all union-closed families whose largest set contains n elements is (1 + o(1))\log_2(n)/(2n) as n approaches infinity. We use a result of Reimer \cite{Reimer03} to show that the density of F is always at least log_2(n)/(2n), verifying Wojcik's conjecture. As a corollary we show that for n \geq 16, some element must appear in at least \sqrt{(\log_2(n))/n}(|F|/2) sets of F.

Motivation & Objective

• To verify Wójcik's conjecture that the minimum density of union-closed families with largest set of size $n$ is asymptotically $(1+o(1))\log_2 n / (2n)$.
• To establish a lower bound on the average set size in union-closed families using Reimer's theorem on the average size of sets in such families.
• To show that for $n \geq 16$, some element appears in at least $\sqrt{(\log_2 n)/n} \cdot |\mathscr{F}|/2$ sets, providing a quantitative strengthening of Frankl's conjecture in a weak sense.
• To provide evidence toward the Frankl conjecture by proving a sharp asymptotic lower bound on the minimum density $s_n$.

Proposed method

• Apply Reimer's inequality: $\frac{1}{|\mathscr{F}|}\sum_{A \in \mathscr{F}} |A| \geq \frac{1}{2}\log_2 |\mathscr{F}|$ to union-closed families.
• Use induction and structural lemmas to show that if $|\mathscr{F}| < n$, then there exist distinct elements $a, b$ with identical set memberships $\mathscr{F}_a = \mathscr{F}_b$.
• Leverage the existence of such duplicate membership sets to derive a lower bound on the total element coverage $\sum_{a \in \bigcup \mathscr{F}} |\mathscr{F}_a|$.
• Combine the lower bound from Reimer's theorem with the structural lemma to derive $D(\mathscr{F}) \geq \log_2 n / (2n)$ for all $n$.
• Use the constructed family $\mathscr{F} = \{A \subseteq \{1,\dots,k\}\} \cup \{\{1,\dots,n\}\}$ with $k = \lceil \log_2 n \rceil$ to show the upper bound $s_n \leq (1+o(1))\log_2 n / (2n)$.
• Derive Corollary 2 by combining the density bound with a case analysis on the minimal set size and using the function $f(x) = 2x^2 / \log_2 x$ to compare bounds.

Experimental results

Research questions

• RQ1 Is the minimum density of union-closed families with largest set of size $n$ asymptotically $\log_2 n / (2n)$ as $n \to \infty$?
• RQ2 Can Reimer's inequality on average set size be used to establish a tight lower bound on the minimum density $s_n$?
• RQ3 Does the existence of duplicate membership sets ($\mathscr{F}_a = \mathscr{F}_b$) in small families allow for improved density bounds?
• RQ4 For $n \geq 16$, is there always an element appearing in at least $\sqrt{(\log_2 n)/n} \cdot |\mathscr{F}|/2$ sets of a union-closed family?
• RQ5 Can the minimum density $s_n$ be bounded both above and below to confirm the asymptotic form $s_n = (1+o(1))\log_2 n / (2n)$?

Key findings

• The minimum density $s_n$ of union-closed families with largest set of size $n$ satisfies $s_n \geq \log_2 n / (2n)$, proven using Reimer's inequality and structural lemmas.
• The asymptotic minimum density is $s_n = (1+o(1))\log_2 n / (2n)$ as $n \to \infty$, confirming Wójcik's conjecture.
• For $n \geq 16$, there exists an element $a$ such that $|\mathscr{F}_a| \geq \frac{1}{2}\sqrt{\frac{\log_2 n}{n}} |\mathscr{F}|$, a quantitative lower bound on element frequency.
• The family $\mathscr{F} = \{A \subseteq \{1,\dots,k\}\} \cup \{\{1,\dots,n\}\}$ with $k = \lceil \log_2 n \rceil$ achieves density $D(\mathscr{F}) = (1+o(1))\log_2 n / (2n)$, showing the bound is tight.
• The proof uses induction and the existence of duplicate membership sets ($\mathscr{F}_a = \mathscr{F}_b$) to derive bounds when $|\mathscr{F}| < n$.
• The method does not prove Frankl's conjecture, as $s_n < 1/2$ for $n \geq 3$, but provides strong evidence for its asymptotic validity.