Remarks on a recent preprint of Chernikov and Towsner
This is an incremental critique that challenges the validity of a specific theorem in model theory and its implications for learning theory, affecting researchers in those fields.
The authors provide a counterexample to a theorem in a recent preprint by Chernikov and Towsner, identify a mistake in its proof, and argue that changes in definitions undermine the paper's claims, particularly its connection to high-arity PAC learning.
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.