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Induced arithmetic removal: complexity 1 patterns over finite fields

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Abstract

We prove an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields. Informally speaking, we show that given a fixed collection of r-colored complexity 1 arithmetic patterns over \({\mathbb{F}_q}\), every coloring \(\phi :\mathbb{F}_q^n\backslash \left\{ 0 \right\} \to [r]\) with o(1) density of every such pattern can be recolored on an o(1)-fraction of the space so that no such pattern remains.

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Acknowledgements

The authors are grateful to Noga Alon, Freddie Manners, and Tom Sanders for helpful discussions, and to the anonymous referee for comments that improved the exposition of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Jacob Fox

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Jonathan Tidor & Yufei Zhao

Authors
  1. Jacob Fox
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  2. Jonathan Tidor
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  3. Yufei Zhao
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Corresponding author

Correspondence to Jonathan Tidor.

Additional information

Fox was supported by a Packard Fellowship and by NSF grant DMS-1855635.

Tidor was supported by NSF Graduate Research Fellowship Program DGE-1745302.

Zhao was supported by NSF Award DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship.

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Fox, J., Tidor, J. & Zhao, Y. Induced arithmetic removal: complexity 1 patterns over finite fields. Isr. J. Math. 248, 1–38 (2022). https://doi.org/10.1007/s11856-022-2290-x

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  • DOI: https://doi.org/10.1007/s11856-022-2290-x

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