Abstract: In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles.\hfill\penalty-10000 \indent \circ \mathcal{P}_{\rm lf,c} (Every locally finite connected graph has a maximal independent set).\hfill\penalty-10000 \indent \circ \mathcal{P}_{\rm lc,c} (Every locally countable connected graph has a maximal independent\hfill\penalty-10000 \indent \quad set).\hfill\penalty-10000 \indent \circ CAC^{\aleph_{\alpha}}_{1} (If in a partially ordered set all antichains are finite and all chains\hfill\penalty-10000 \indent \quad have size \aleph_{\alpha}, then the set has size \aleph_{\alpha}) if \aleph_{\alpha} is regular.\hfill\penalty-10000 \indent \circ CWF (Every partially ordered set has a cofinal well-founded subset).\hfill\penalty-10000 \indent \circ \mathcal{P}_{G,H_{2}} (For any infinite graph G=(V_{G}, E_{G}) and any finite graph H=\hfill\penalty-10000 \indent \quad (V_{H}, E_{H}) on 2 vertices, if every finite subgraph of G has a homomorphism\hfill\penalty-10000 \indent \quad into H, then so has G).\hfill\penalty-10000 \indent \circ If G=(V_{G},E_{G}) is a connected locally finite chordal graph, then there is\hfill\penalty-10000 \indent \quad an ordering ``<" of V_{G} such that \{w < v \colon \{w,v\} \in E_{G}\} is a clique for each\hfill\penalty-10000 \indent \quad v\in V_{G}.
Keywords: variants of chain/antichain principle; graph homomorphism; maximal independent sets; cofinal well-founded subsets of partially ordered sets; axiom of choice; Fraenkel-Mostowski (FM) permutation models of ZFA + \neg AC
DOI: DOI 10.14712/1213-7243.2023.028
AMS Subject Classification: 03E25 03E35 06A07 05C69