On the maximal size of $(a,b)$-town$\pmod k$ families
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| Format: | Preprint |
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2025
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| _version_ | 1866918152262123520 |
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| author | Veselinov, Nikola Marinov, Miroslav |
| author_facet | Veselinov, Nikola Marinov, Miroslav |
| contents | A family $\mathcal{F}\subseteq\mathcal{P}(n)$ is an $(a,b)$-town$\pmod k$ if all sets in it have cardinality $a\pmod k$ and all pairwise intersections in it have cardinality $b\pmod k$. For $k=2$ the maximal size of such a family is known for each $a,b$, while for $k=3$ only $b-a\equiv 2 \pmod 3$ is fully understood. We provide a bound for $k=3$ when $b-a\equiv 1 \pmod 3$ and $n\equiv 2 \pmod 3$, which turns out to be tight for infinitely many such $n$. We also give sufficient conditions on the parameters $a,b,k,n$, which result in a better bound than the one from general settings by Ray-Chaudhuri--Wilson, in particular showing that this bound occurs infinitely often in a sense where all of $a,b,n$ can vary for a fixed $k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the maximal size of $(a,b)$-town$\pmod k$ families Veselinov, Nikola Marinov, Miroslav Combinatorics 05D05 (Primary), 05E99 (Secondary) A family $\mathcal{F}\subseteq\mathcal{P}(n)$ is an $(a,b)$-town$\pmod k$ if all sets in it have cardinality $a\pmod k$ and all pairwise intersections in it have cardinality $b\pmod k$. For $k=2$ the maximal size of such a family is known for each $a,b$, while for $k=3$ only $b-a\equiv 2 \pmod 3$ is fully understood. We provide a bound for $k=3$ when $b-a\equiv 1 \pmod 3$ and $n\equiv 2 \pmod 3$, which turns out to be tight for infinitely many such $n$. We also give sufficient conditions on the parameters $a,b,k,n$, which result in a better bound than the one from general settings by Ray-Chaudhuri--Wilson, in particular showing that this bound occurs infinitely often in a sense where all of $a,b,n$ can vary for a fixed $k$. |
| title | On the maximal size of $(a,b)$-town$\pmod k$ families |
| topic | Combinatorics 05D05 (Primary), 05E99 (Secondary) |
| url | https://arxiv.org/abs/2510.00251 |