Two new results on maximal left-compressed intersecting families
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909900731318272 |
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| author | Flower, Allan Mycroft, Richard |
| author_facet | Flower, Allan Mycroft, Richard |
| contents | This paper presents two new results on the theory of maximal left-compressed intersecting families (MLCIFs). First, we answer a question raised by Barber by showing that the number of $k$-uniform MLCIFs on a ground set of size $n$ grows as a doubly-exponential function of $k$, which we identify up to a log factor in the exponent. Among these MLCIFs we identify $k$ specific MLCIFs -- which we call the canonical MLCIFs -- as being in a meaningful way the most important MLCIFs. Specifically, our second main result shows that the canonical MLCIFs are precisely those which can have maximum weight among all $k$-uniform MLCIFs under a non-trivial increasing weight function, and moreover that each canonical MLCIF is the unique $k$-uniform MLCIF of maximum weight for some increasing weight function. This gives an interesting generalisation of the Erdős--Ko--Rado theorem to a notion of size which places greater significance on some elements of the ground set than others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10592 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two new results on maximal left-compressed intersecting families Flower, Allan Mycroft, Richard Combinatorics This paper presents two new results on the theory of maximal left-compressed intersecting families (MLCIFs). First, we answer a question raised by Barber by showing that the number of $k$-uniform MLCIFs on a ground set of size $n$ grows as a doubly-exponential function of $k$, which we identify up to a log factor in the exponent. Among these MLCIFs we identify $k$ specific MLCIFs -- which we call the canonical MLCIFs -- as being in a meaningful way the most important MLCIFs. Specifically, our second main result shows that the canonical MLCIFs are precisely those which can have maximum weight among all $k$-uniform MLCIFs under a non-trivial increasing weight function, and moreover that each canonical MLCIF is the unique $k$-uniform MLCIF of maximum weight for some increasing weight function. This gives an interesting generalisation of the Erdős--Ko--Rado theorem to a notion of size which places greater significance on some elements of the ground set than others. |
| title | Two new results on maximal left-compressed intersecting families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.10592 |