Induced arithmetic removal for partition-regular patterns of complexity 1

TL;DR

This paper proves that all partition-regular arithmetic patterns of complexity 1 can be completely removed from vector spaces over finite fields through appropriate recoloring, extending previous results on pattern elimination.

Contribution

It establishes that complete pattern removal is possible for all partition-regular arithmetic patterns of complexity 1, generalizing prior partial results.

Findings

Complete removal of all partition-regular arithmetic patterns of complexity 1.

Extension of pattern elimination results to all such patterns.

Recoloring methods guarantee pattern-free spaces with no exceptions.

Abstract

In 2019, Fox, Tidor and Zhao (arXiv:1911.03427) proved an induced arithmetic removal lemma for linear patterns of complexity 1 in vector spaces over a fixed finite field. With no further assumptions on the pattern, this induced removal lemma cannot guarantee a fully pattern-free recolouring of the space, as some `non-generic' instances must necessarily remain. On the other hand, Bhattacharyya et al. (arXiv:1212.3849) showed that in the case of translation-invariant patterns, it is possible to obtain recolourings that eliminate the given pattern completely, with no exceptions left behind. This paper demonstrates that such complete removal can be achieved for all partition-regular arithmetic patterns of complexity 1.