Remarks on a recent preprint of Chernikov and Towsner

TL;DR

The note analyzes two versions of Chernikov and Towsner's preprint, showing that v1's Theorem 6.4 is false via a concrete counterexample and that relying on finite slicewise VC dimension is essential to avoid such failures. It then critiques v2 by highlighting that the revised learning notion (KT) is misaligned with high-arity PAC learning and CM25b, and that the proposed proof contains elementary mistakes, notably centers depending on an unknown measure. Together, these points clarify the boundaries between $k$-dependence and high-arity PAC learning and emphasize the need to align definitions to preserve connections with CM25a/CM25b rather than accept the revised claims. The discussion cautions against relying on unbounded-slice learning in finite spaces and shows how a seemingly minor definitional shift can disrupt established correspondences. The overall message is a careful delineation of where the Chernikov-Towsner results intersect or diverge from Coregliano-Malliaris frameworks.

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.