A Bollobás-type problem: from root systems to Erdős-Ko-Rado
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910401882488832 |
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| author | Browne, Patrick J. Gashi, Qëndrim R. Catháin, Padraig Ó |
| author_facet | Browne, Patrick J. Gashi, Qëndrim R. Catháin, Padraig Ó |
| contents | Motivated by an Erdős--Ko--Rado type problem on sets of strongly orthogonal roots in the $A_{\ell}$ root system, we estimate bounds for the size of a family of pairs $(A_{i}, B_{i})$ of $k$-subsets in $\{ 1, 2, \ldots, n\}$ such that $A_{i} \cap B_{j}= \emptyset$ and $|A_{i} \cap A_{j}| + |B_{i} \cap B_{j}| = k$ for all $i \neq j$. This is reminiscent of a classic problem of Bollobás. We provide upper and lower bounds for this problem, relying on classical results of extremal combinatorics and an explicit construction using the incidence matrix of a finite projective plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Bollobás-type problem: from root systems to Erdős-Ko-Rado Browne, Patrick J. Gashi, Qëndrim R. Catháin, Padraig Ó Combinatorics 05D05, 17B22 Motivated by an Erdős--Ko--Rado type problem on sets of strongly orthogonal roots in the $A_{\ell}$ root system, we estimate bounds for the size of a family of pairs $(A_{i}, B_{i})$ of $k$-subsets in $\{ 1, 2, \ldots, n\}$ such that $A_{i} \cap B_{j}= \emptyset$ and $|A_{i} \cap A_{j}| + |B_{i} \cap B_{j}| = k$ for all $i \neq j$. This is reminiscent of a classic problem of Bollobás. We provide upper and lower bounds for this problem, relying on classical results of extremal combinatorics and an explicit construction using the incidence matrix of a finite projective plane. |
| title | A Bollobás-type problem: from root systems to Erdős-Ko-Rado |
| topic | Combinatorics 05D05, 17B22 |
| url | https://arxiv.org/abs/2404.04867 |