A Note on the Union-closed Sets Conjecture

Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros. Specifically, the existence of such columns when the rows of the matrix are closed under $\textit{material conditional}$ leads us to a weaker version of the $\textit{Union-Closed Set Conjecture}$.Comment: 11 page

A note on the union-closed sets conjecture

A collection A of finite sets is closed under union if A, B ∈ A implies that A ∪ B ∈ A. The Union-Closed Sets Conjecture states that if A is a union-closed collection of sets, containing at least one non-empty set, then there is an element which belongs to at least half of the sets in A. We show that if q is the minimum cardinality of ∪A taken over allcounterexamples A, then any counterexample A has cardinality at least 4q − 1

The number of abundant elements in union-closed families without small sets

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical Frankl's conjecture (1979) asserts that $\mathcal{F}$ has an abundant element. As possible strengthenings, Poonen (1992) conjectured that if $\mathcal{F}$ has precisely one abundant element, then this element belongs to each set of $\mathcal{F}$, and Cui and Hu (2019) investigated whether $\mathcal{F}$ has at least $k$ abundant elements if a smallest set of $\mathcal{F}$ is of size at least $k$. Cui and Hu conjectured that this holds for $k = 2$ and asked whether this also holds for the cases $k = 3$ and $k > \frac{n}{2}$ where $n$ is the size of the largest set of $\mathcal{F}$. We show that $\mathcal{F}$ has at least $k$ abundant elements if $k \geq n - 3$, and that $\mathcal{F}$ has at least $k - 1$ abundant elements if $k = n - 4$, and we construct a union-closed family with precisely $k - 1$ abundant elements for every $k$ and $n$ satisfying $n - 4 \geq k \geq 3$ and $n \geq 9$ (and for $k = 3$ and $n = 8$). We also note that $\mathcal{F}$ always has at least $\min \{ n, 2k - n + 1 \}$ abundant elements. On the other hand, we construct a union-closed family with precisely two abundant elements for every $k$ and $n$ satisfying $n \geq \max \{ 3, 5k-4 \}$. Lastly, we show that Cui and Hu's conjecture for $k = 2$ stands between Frankl's conjecture and Poonen's conjecture

FC-families, and improved bounds for Frankl's Conjecture

A family of sets F is said to be union-closed if A \cup B is in F for every A and B in F. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the sets. Here we prove that the conjecture holds for families containing three 3-subsets of a 5-set, four 3-subsets of a 6-set, or eight 4-subsets of a 6-set, extending work of Poonen and Vaughan. As an application we prove the conjecture in the case that the largest set has at most nine elements, extending a result of Gao and Yu. We also pose several open questions.Comment: 19 pgs, no figure