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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.00138 |
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| _version_ | 1866915470752350208 |
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| author | Csizmadia, Miklós |
| author_facet | Csizmadia, Miklós |
| contents | Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there exists three concurrent lines in the plane such that the union of the three lines intersect every member of $F$. This had previously only been proven for lines that are not required to be concurrent by McGinnis and Zerbib in arXiv:2103.05565v2. In fact, I prove a more general, ``colorful'' version of this result: If $F_1, \dots , F_5$ are bounded families of compact connected sets in the plane such that every three sets, chosen from three distinct families $F_i$, can be pierced by a single line, then there exists $1 \leq j \leq 5$ and three concurrent lines, such that the union of the three lines intersect every member of $F_j$. McGinnis and Zerbib had 6 families instead of 5, so I also improve their result in this respect. Moreover, the result can also be extended to unbounded families, if we allow the piercing lines to be parallel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00138 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improvement on line transversals of families of connected sets in the plane Csizmadia, Miklós Combinatorics Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there exists three concurrent lines in the plane such that the union of the three lines intersect every member of $F$. This had previously only been proven for lines that are not required to be concurrent by McGinnis and Zerbib in arXiv:2103.05565v2. In fact, I prove a more general, ``colorful'' version of this result: If $F_1, \dots , F_5$ are bounded families of compact connected sets in the plane such that every three sets, chosen from three distinct families $F_i$, can be pierced by a single line, then there exists $1 \leq j \leq 5$ and three concurrent lines, such that the union of the three lines intersect every member of $F_j$. McGinnis and Zerbib had 6 families instead of 5, so I also improve their result in this respect. Moreover, the result can also be extended to unbounded families, if we allow the piercing lines to be parallel. |
| title | Improvement on line transversals of families of connected sets in the plane |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.00138 |