Abstract
The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead prove that $2a$ is an upper bound to the number of sets in a union-closed family on a ground set of $n$ elements where each element is in at most $a$ sets for all $a,n\in \mathbb{N}^+$. Similarly, one could prove that the minimum number of sets containing the most frequent element in a (non-empty) union-closed family with $m$ sets and $n$ elements is at least $\frac{m}{2}$ for any $m,n\in \mathbb{N}^+$. Formulating these problems as integer programs, we observe that the optimal values we computed do not vary with $n$. We formalize these observations as conjectures, and show that they are not equivalent to the Frankl conjecture while still having wide-reaching implications if proven true. Finally, we prove special cases of the new conjectures and discuss possible approaches to solve them completely.
Highlights
The union-closed sets conjecture is a celebrated open problem in combinatorics which was popularized by Frankl in the late 1970’s [Fra83], and is often referred to as the Frankl conjecture
Instead of proving the Frankl conjecture, one could instead try to prove that any union-closed family contains an element present in at least some fraction of the sets, just as Knill did in the following theorem
Recall that Conjecture 11 is equivalent to the Frankl conjecture, and so Theorem 16 could be reformulated in such terms
Summary
The union-closed sets conjecture is a celebrated open problem in combinatorics which was popularized by Frankl in the late 1970’s [Fra83], and is often referred to as the Frankl conjecture. In a union-closed family F such that S(F) = {∅}, there exists an element of E(F) that is in at least half of the sets of S(F). Let me(F) be the number of sets in F containing some element e ∈ E(F). Let the degree of F, denoted by a(F), be the maximum number of sets in F containing any element of E(F), that is, a(F) =. Let e∗(F ) be an arbitrary element of maximum degree, i.e., any of possibly many elements in E(F) contained in a(F) sets.
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