  In set theory without the Axiom of Choice (AC), we observe new relations of
the following statements with weak choice principles. 1. Every locally finite
connected graph has a maximal independent set. 2. Every locally countable
connected graph has a maximal independent set. 3. 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. 4. Every
partially 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 an
ordering $<$ of $V_{G}$ such that $\{w < v : \{w,v\} \in E_{G}\}$ is a clique
for each $v\in V_{G}$.
