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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.00586 |
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| _version_ | 1866912562331779072 |
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| author | Bukh, Boris Chao, Ting-Wei Zheng, Zeyu |
| author_facet | Bukh, Boris Chao, Ting-Wei Zheng, Zeyu |
| contents | A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when is a $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. For composite moduli with $ω$ distinct prime factors, the argument of Szegedy gives an upper bound of $ωn-ω\log_2 n$ on the size of an $\ell$-Oddtown. We improve this to $ωn-(2ω+\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Oddtown problem modulo a composite number Bukh, Boris Chao, Ting-Wei Zheng, Zeyu Combinatorics 05D05, 05B20 A family of subsets $\mathcal{A}$ of an $n$-element set is called an $\ell$-Oddtown if the sizes of all sets are not divisible by $\ell$, but the sizes of pairwise intersections are divisible by $\ell$. Berlekamp and Graver showed that when is a $\ell$ is a prime, the maximum size of an $\ell$-Oddtown is $n$. For composite moduli with $ω$ distinct prime factors, the argument of Szegedy gives an upper bound of $ωn-ω\log_2 n$ on the size of an $\ell$-Oddtown. We improve this to $ωn-(2ω+\varepsilon)\log_2 n$ for most $\ell$ and $n$ using a combination of linear algebraic and Fourier-analytic arguments. |
| title | The Oddtown problem modulo a composite number |
| topic | Combinatorics 05D05, 05B20 |
| url | https://arxiv.org/abs/2509.00586 |