Average-Rare Order Ideals in Functional Preorders

TL;DR

This paper proves that the family of order ideals in certain functional preorders has a nonpositive normalized degree sum, introduces a conjecture related to Frankl's Conjecture, and verifies proofs using Lean 4.

Contribution

It establishes the average-rare property for order ideals in functional preorders and rooted forests, and proposes a related conjecture, with formal verification in Lean 4.

Findings

Order ideals in functional preorders are average-rare.

The same property holds for rooted forests.

A new conjecture related to Frankl's Conjecture is proposed.

Abstract

We prove that for the preorder induced by a function f: V -> V, the family of all order ideals is average-rare, that is, its normalized degree sum (nds) is nonpositive. As a base case in our reduction, we establish the same result for functional partial orders (or rooted forests). We also propose a conjecture related to Frankl's Conjecture. All proofs have been formally verified in the proof assistant Lean 4.