A constant lower bound for the union-closed sets conjecture

The union-closed sets conjecture has been a longstanding open problem in combinatorics. In this paper, the authors provide a constant lower bound for the conjecture, improving upon previous bounds. Specifically, they show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This result follows from an information theoretic strengthening of the conjecture. The authors introduce a lemma that provides an upper bound on the entropy of a set given its frequency within a family. They then use this lemma to prove their main result by showing that if $A$ and $B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \cup B) > H(A)$. In addition to providing this new lower bound, the authors also discuss potential avenues for further progress on the conjecture. They suggest studying the KL-divergence between distributions as a way to quantify how far from uniform the distribution of $A\cup B$ may be when $Pr[i\in A]$ is close to $\frac12$. They propose Conjecture 1 as a possible path towards resolving the union-closed sets conjecture: Let A, B be iid samples from a distribution over a family of subsets of [n]. Assume that P r[i ∈ A] < 0.5 for all i, and H(A) > 0. Then H(A ∪ B) + D(A ∪ B||A) > H(A). Overall, this paper provides a significant contribution to the study of union-closed families and offers new insights into potential directions for future research.

The Union-Closed Sets Conjecture: A Constant Lower Bound

The union-closed sets conjecture has been a longstanding open problem in combinatorics. In this paper, the authors provide a constant lower bound for the conjecture, improving upon previous bounds. Specifically, they show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This result follows from an information theoretic strengthening of the conjecture.

Information Theoretic Strengthening of the Conjecture

The authors introduce a lemma that provides an upper bound on the entropy of a set given its frequency within a family. They then use this lemma to prove their main result by showing that if $A$ and $B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \cup B) > H(A)$.

Constant Lower Bound

This new lower bound shows that for any union-closed family $\mathcal{F}\subseteq 2^{[n]}$, there exists an element i ∈ [n] which is contained in at least 0.01 fraction of sets in F . This improves upon previous bounds as it provides a constant lower bound rather than one dependent on n or |F| .

Further Progress on the Conjecture

In addition to providing this new lower bound, the authors also discuss potential avenues for further progress on the conjecture. They suggest studying the KL-divergence between distributions as a way to quantify how far from uniform the distribution of A∪B may be when Pr[i∈A] is close to 1/2 . They propose Conjecture 1 as a possible path towards resolving the union-closed sets conjecture: Let A, B be iid samples from a distribution over a family of subsets of [n]. Assume that P r[i ∈ A] < 0.5 for all i, and H(A) > 0 . Then H(A ∪ B) + D(A ∪ B||A) > H(A).

Conclusion

Overall, this paper provides a significant contribution to the study of union-closed families and offers new insights into potential directions for future research.