Color-detectors of hypergraphs
Let \(X\) be a set of cardinality \(k\), \(\mathcal{F}\) be a family of subsets of \(X\). We say that a cardinal \(\lambda, \lambda< k\), is a color-detector of the hypergraph \(H=(X, \mathcal{F})\) if card \(\chi(X)\leq \lambda\) for every coloring \(\chi: X\rightarrow k\) such that card ...
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| Datum: | 2018 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/918 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | Let \(X\) be a set of cardinality \(k\), \(\mathcal{F}\) be a family of subsets of \(X\). We say that a cardinal \(\lambda, \lambda< k\), is a color-detector of the hypergraph \(H=(X, \mathcal{F})\) if card \(\chi(X)\leq \lambda\) for every coloring \(\chi: X\rightarrow k\) such that card \(\chi(F)\leq \lambda\) for every \(F\in \mathcal{F}\). We show that the color-detectors of \(H\) are tightly connected with the covering number \(cov (H)=\sup \{\alpha: \text{ any } \alpha \text{ points of } X \text{ are contained in some }\ F\in \mathcal{F} \}\). In some cases we determine all of the color-detectors of \(H\) and their asymptotic counterparts. We put also some open questions. |
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