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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.13588 |
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| _version_ | 1866909552660709376 |
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| author | Johnston, Griffin O'Neill, Jason |
| author_facet | Johnston, Griffin O'Neill, Jason |
| contents | Given a vector $α= (α_1, \ldots, α_k) \in \mathbb{F}_2^k$, we say a collection of subsets $\mathcal{F}$ satisfies $α$-intersection pattern modulo $2$ if all $i$-wise intersections consisting of $i$ distinct sets from $\mathcal{F}$ have size $α_i \pmod{2}$. In this language, the classical oddtown and eventown problems correspond to vectors $α=(1,0)$ and $α=(0,0)$ respectively. In this paper, we determine the largest such set families of subsets on a $n$-element set with $α$-intersection pattern modulo $2$ for all $α\in \mathbb{F}_2^3$ and all $α\in \mathbb{F}_2^4$ asymptotically. Lastly, we consider the corresponding problem with restrictions modulo $3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13588 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A few new oddtown and eventown problems Johnston, Griffin O'Neill, Jason Combinatorics Given a vector $α= (α_1, \ldots, α_k) \in \mathbb{F}_2^k$, we say a collection of subsets $\mathcal{F}$ satisfies $α$-intersection pattern modulo $2$ if all $i$-wise intersections consisting of $i$ distinct sets from $\mathcal{F}$ have size $α_i \pmod{2}$. In this language, the classical oddtown and eventown problems correspond to vectors $α=(1,0)$ and $α=(0,0)$ respectively. In this paper, we determine the largest such set families of subsets on a $n$-element set with $α$-intersection pattern modulo $2$ for all $α\in \mathbb{F}_2^3$ and all $α\in \mathbb{F}_2^4$ asymptotically. Lastly, we consider the corresponding problem with restrictions modulo $3$. |
| title | A few new oddtown and eventown problems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.13588 |