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arXiv · 2507.22831

On a Ramsey--Tur\'{a}n variant of Roth's theorem

Matija Bucić·Micha Christoph·Jaehoon Kim·Hyunwoo Lee·Varun Sivashankar
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Abstract

A classical theorem of Roth states that the maximum size of a solution-free set of a homogeneous linear equation $\mathcal{L}$ in $\mathbb{F}_p$ is $o(p)$ if and only if the sum of the coefficients of $\mathcal{L}$ is $0$. In this paper, we prove a Ramsey--Tur\'{a}n variant of Roth's theorem, with respect to a natural notion of ``structured'' sets introduced by Erd\H{o}s and S\'ark\"ozy in the 1970's. Namely, we show that the following statements are equivalent: $(a)$ Every solution-free set $A$ of $\mathcal{L}$ in $\mathbb{F}_p$ with $\alpha(\mathrm{Cay}_{\mathbb{F}_p}(A)) = o(p)$ has size $o(p)$. $(b)$ There exists a non-empty \emph{subset} of coefficients of $\mathcal{L}$ with zero sum.

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Matija Bucić, Micha Christoph, Jaehoon Kim, Hyunwoo Lee, Varun Sivashankar. 2025-07-30. On a Ramsey--Tur\'{a}n variant of Roth's theorem. https://arxiv.org/abs/2507.22831

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