Remarks on Frankl's conjecture

TL;DR

This paper explores reformulations of Frankl's conjecture using various mathematical structures and proposes a stronger, potentially more approachable conjecture, supported by an inequality that could aid in its proof.

Contribution

It introduces a stronger conjecture related to Frankl's conjecture and proves an inequality that may facilitate its proof.

Findings

Reformulations of Frankl's conjecture in terms of families, matrices, and lattices

Proposal of a stronger conjecture with a clearer formulation

Proof of an inequality that supports the stronger conjecture

Abstract

First a few reformulations of Frankl's conjecture are given, in terms of reduced families or matrices, or analogously in terms of lattices. These lead naturally to a stronger conjecture with a neat formulation which might be easier to attack than Frankl's. To this end we prove an inequality which might help in proving the stronger conjecture.