Cross-free families have linear size
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | |
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| _version_ | 1866918373246369792 |
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| author | Tomon, István |
| author_facet | Tomon, István |
| contents | Two subsets $A$ and $B$ of a ground set $X$ are \emph{crossing} if none of the four sets $A\setminus B,B\setminus A,A\cap B, X\setminus (A\cup B)$ are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an $n$-element ground set with no $k$-pairwise crossing members has size $O(kn)$. We prove the bound $O_k(n)$, settling (arguably) the main problem about the growth rate of such families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05443 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cross-free families have linear size Tomon, István Combinatorics Two subsets $A$ and $B$ of a ground set $X$ are \emph{crossing} if none of the four sets $A\setminus B,B\setminus A,A\cap B, X\setminus (A\cup B)$ are empty. Almost fifty years ago, Karzanov and Lomonosov conjectured that every family of subsets of an $n$-element ground set with no $k$-pairwise crossing members has size $O(kn)$. We prove the bound $O_k(n)$, settling (arguably) the main problem about the growth rate of such families. |
| title | Cross-free families have linear size |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.05443 |