On supratopologies, normalized families and Frankl conjecture
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909791468650496 |
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| author | Carvalho, André Machiavelo, António |
| author_facet | Carvalho, André Machiavelo, António |
| contents | We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the structure of normalized families, presenting a new way of reducing such a family to a smaller one using dual families. Applying our reduction method, we prove a refinement of a conjecture originally proposed by Poonen. Finally, we show that Frankl's Conjecture holds for the class of families obtained from successively applying the reduction process to a power set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On supratopologies, normalized families and Frankl conjecture Carvalho, André Machiavelo, António Combinatorics We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the structure of normalized families, presenting a new way of reducing such a family to a smaller one using dual families. Applying our reduction method, we prove a refinement of a conjecture originally proposed by Poonen. Finally, we show that Frankl's Conjecture holds for the class of families obtained from successively applying the reduction process to a power set. |
| title | On supratopologies, normalized families and Frankl conjecture |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.11213 |