A Sauer-Shelah-Perles Lemma for Lattices

TL;DR

This paper explores lattice-theoretical extensions of the Sauer-Shelah-Perles Lemma, proposing a conjecture linking its validity to the property of being relatively complemented, and provides partial proofs towards this conjecture.

Contribution

It introduces a conjecture that the Sauer-Shelah-Perles Lemma extends to lattices if and only if they are relatively complemented, and offers partial results supporting this claim.

Findings

Partial results supporting the conjecture

Proposed a characterization of lattices satisfying the lemma

Established connections between lattice properties and combinatorial bounds

Abstract

We study lattice-theoretical extensions of the celebrated Sauer-Shelah-Perles Lemma. We conjecture that a general Sauer-Shelah-Perlem Lemma holds for a lattice $L$ if and only if $L$ is relatively complemented, and prove partial results towards this conjecture.