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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.06521 |
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| _version_ | 1866910110275600384 |
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| author | Ivan, Maria-Romina Jaffe, Sean |
| author_facet | Ivan, Maria-Romina Jaffe, Sean |
| contents | What is the smallest size of a family of subsets of $[n]$ such that it does not contain an induced copy of $Q_2$ as a poset (known as the \textit{diamond}), but adding a new set creates such a copy? It is easy to see that a maximal chain has this property, and thus the answer is at most $n+1$. Despite the simplicity of the diamond structure, the lower bound stagnated at $\sqrt n$ for quite some time, until recently the authors obtained a linear lower bound. In this paper, we fully solve this question showing that such a family must have size at least $n+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06521 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Exact Saturation Number for the Diamond Ivan, Maria-Romina Jaffe, Sean Combinatorics What is the smallest size of a family of subsets of $[n]$ such that it does not contain an induced copy of $Q_2$ as a poset (known as the \textit{diamond}), but adding a new set creates such a copy? It is easy to see that a maximal chain has this property, and thus the answer is at most $n+1$. Despite the simplicity of the diamond structure, the lower bound stagnated at $\sqrt n$ for quite some time, until recently the authors obtained a linear lower bound. In this paper, we fully solve this question showing that such a family must have size at least $n+1$. |
| title | The Exact Saturation Number for the Diamond |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.06521 |