Article
| Title: | Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC (English) |
| Author: | Banerjee, Amitayu |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 64 |
| Issue: | 2 |
| Year: | 2023 |
| Pages: | 137-159 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ $\mathcal{P}_{\rm lf,c}$ (Every locally finite connected graph has a maximal independent set). $\circ$ $\mathcal{P}_{\rm lc,c}$ (Every locally countable connected graph has a maximal independent set). $\circ$ CAC$^{\aleph_{\alpha}}_{1}$ (If in a partially ordered set all antichains are finite and all chains have size $\aleph_{\alpha}$, then the set has size $\aleph_{\alpha}$) if $\aleph_{\alpha}$ is regular. $\circ$ CWF (Every partially ordered set has a cofinal well-founded subset). $\circ$ $\mathcal{P}_{G,H_{2}} $ (For any infinite graph $ G=(V_{G}, E_{G}) $ and any finite graph $ H=(V_{H}, E_{H})$ on 2 vertices, if every finite subgraph of $G$ has a homomorphism into $H$, then so has $G$). $\circ$ If $ G=(V_{G},E_{G}) $ is a connected locally finite chordal graph, then there is an ordering ``$<$" of $V_{G}$ such that $\{w < v \colon \{w,v\} \in E_{G}\}$ is a clique for each $v\in V_{G}$. (English) |
| Keyword: | variants of chain/antichain principle |
| Keyword: | graph homomorphism |
| Keyword: | maximal independent sets |
| Keyword: | cofinal well-founded subsets of partially ordered sets |
| Keyword: | axiom of choice |
| Keyword: | Fraenkel--Mostowski (FM) permutation models of ZFA + $\neg$ AC |
| MSC: | 03E25 |
| MSC: | 03E35 |
| MSC: | 05C69 |
| MSC: | 06A07 |
| idZBL: | Zbl 07790588 |
| idMR: | MR4658996 |
| DOI: | 10.14712/1213-7243.2023.028 |
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| Date available: | 2023-12-13T13:31:57Z |
| Last updated: | 2024-02-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151857 |
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