Remarks on a recent preprint of Chernikov and Towsner

TL;DR

This paper critiques a recent preprint by Chernikov and Towsner, providing a counterexample, identifying a mistake in their proof, and explaining how their revised theorem affects the connection to high-arity PAC learning.

Contribution

It offers a counterexample to a key theorem, highlights a proof mistake, and clarifies the impact of definitional changes on their theoretical claims.

Findings

Counterexample disproves the original theorem

Identification of a mistake in the proof

Revised theorem alters the connection to PAC learning

Abstract

In this brief note, we first give a counterexample to a theorem in Chernikov and Towsner, arXiv:2510.02420(1). In arXiv:2510.02420(2), the theorem has changed but as we explain the proof has a mistake. The change in the statement, due to changes in the underlying definition, affects the paper's claims. Since that theorem had been relevant to connecting the work of their paper to Coregliano-Malliaris high-arity PAC learning, a connection which now disappears, we also explain why their definitions miss crucial aspects that our work was designed to grapple with.