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Induced arithmetic removal for partition-regular patterns of complexity 1

V. Gladkova

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.

Induced arithmetic removal for partition-regular patterns of complexity 1

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.

Paper Structure

This paper contains 7 sections, 16 theorems, 21 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview of the argument
  4. Linear systems and partition regularity
  5. Proof of Theorem \ref{['main']}
  6. Discussion
  7. Proof of the counting lemma

Key Result

Theorem 1.2

Fix $\epsilon > 0$, an integer $r > 0$, and an arithmetic pattern $\mathcal{H}$ of complexity 1. There exists a $\delta = \delta(\epsilon, r, \mathcal{H})$ satisfying the following. If $\phi:\mathbb{F}_{p}^{n} \rightarrow [r]$ is an $r$-colouring of $\mathbb{F}_{p}^{n}$ such that $\Lambda_{\mathcal{

Figures (5)

  • Figure 1: The sets $A$ (left) and $A'$ (right) depicted as hashed areas. Blue squares correspond to regular cosets after an application of the arithmetic regularity lemma.
  • Figure 2: An instance of $\mathcal{L}$ in $A'$ would imply many instances of $\mathcal{L}$ in $A$ as a consequence of Lemma \ref{['lemma:counting-lemma']}.
  • Figure 3: The strategy of Fox, Tidor, and Zhao induced-1 for the induced arithmetic removal of complexity-1 patterns, depicted for a 2-colouring. Instances of the pattern which contain 0 may remain.
  • Figure 4: An auxiliary colouring $\psi$ arising in the proof of Lemma \ref{['lemma:third-partition']} from a 3-colouring $\phi:\mathbb{F}_{p}^{n} \rightarrow \{r,b,g\}$. Due to a large proportion of regular cosets (shown in blue), we can find a monochromatic solution consisting entirely of regular cosets.
  • Figure :

Theorems & Definitions (37)

  • Definition 1.1: Arithmetic patterns
  • Theorem 1.2: Induced removal for complexity 1 patterns induced-1
  • Definition 1.3: Partition regularity
  • Example 1.4
  • Theorem 1.5: Induced removal for partition-regular patterns
  • Example 1.6
  • Corollary 1.7
  • proof
  • Definition 2.1: Pattern density
  • Definition 2.2: Fourier uniformity
  • ...and 27 more