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In set theory without the Axiom of Choice (AC), we observe new relations ofthe following statements with weak choice principles. 1. Every locally finiteconnected graph has a maximal independent set. 2. Every locally countableconnected graph has a maximal independent set. 3. If in a partially ordered setall antichains are finite and all chains have size $\aleph_{\alpha}$, then theset has size $\aleph_{\alpha}$ if $\aleph_{\alpha}$ is regular. 4. Everypartially ordered set has a cofinal well-founded subset. 5. If$G=(V_{G},E_{G})$ is a connected locally finite chordal graph, then there is anordering $<$ of $V_{G}$ such that $\{w < v : \{w,v\} \in E_{G}\}$ is a cliquefor each $v\in V_{G}$.

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC