Induced arithmetic removal: complexity 1 patterns over finite fields

11/08/2019

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by   Jacob Fox, et al.

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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 F_q, every coloring ϕF_q^n ∖{0}→ [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.