On a Ramsey--Turán variant of Roth's theorem
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916871398227968 |
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| author | Bucić, Matija Christoph, Micha Kim, Jaehoon Lee, Hyunwoo Sivashankar, Varun |
| author_facet | Bucić, Matija Christoph, Micha Kim, Jaehoon Lee, Hyunwoo Sivashankar, Varun |
| contents | 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--Turán variant of Roth's theorem, with respect to a natural notion of ``structured'' sets introduced by Erdős and Sárközy 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 $α(\mathrm{Cay}_{\mathbb{F}_p}(A)) = o(p)$ has size $o(p)$.
$(b)$ There exists a non-empty \emph{subset} of coefficients of $\mathcal{L}$ with zero sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Ramsey--Turán variant of Roth's theorem Bucić, Matija Christoph, Micha Kim, Jaehoon Lee, Hyunwoo Sivashankar, Varun Combinatorics 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--Turán variant of Roth's theorem, with respect to a natural notion of ``structured'' sets introduced by Erdős and Sárközy 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 $α(\mathrm{Cay}_{\mathbb{F}_p}(A)) = o(p)$ has size $o(p)$. $(b)$ There exists a non-empty \emph{subset} of coefficients of $\mathcal{L}$ with zero sum. |
| title | On a Ramsey--Turán variant of Roth's theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.22831 |