Roth-type theorems in $K_{s,t}$-free sets
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866912849986584576 |
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| author | Jing, Yifan Pohoata, Cosmin Xu, Max Wenqiang |
| author_facet | Jing, Yifan Pohoata, Cosmin Xu, Max Wenqiang |
| contents | We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $Ω(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets.
We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lovász-Sauermann. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18738 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Roth-type theorems in $K_{s,t}$-free sets Jing, Yifan Pohoata, Cosmin Xu, Max Wenqiang Combinatorics Number Theory We show that for all integers $2\le s\le t$, any $K_{s,t}$-free subset of $[N]$ with size $Ω(n^{1-1/s})$ must contain a nontrivial solution to every fixed translation-invariant linear equation in at least five variables. This extends earlier results for Sidon sets due to Conlon-Fox-Sudakov-Zhao and Prendiville to the full family of $K_{s,t}$-free sets. We also study the corresponding problem in vector spaces over finite fields. In $\mathbb F_q^n$ we obtain stronger quantitative bounds, including polylogarithmic savings, by combining Fourier-analytic transference with polynomial-method input from the arithmetic cycle-removal lemma of Fox-Lovász-Sauermann. |
| title | Roth-type theorems in $K_{s,t}$-free sets |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2601.18738 |