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In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ ${𝒫}_{\mathrm{lf},\mathrm{c}}$ (Every locally finite connected graph has a maximal independent set). $\circ$ ${𝒫}_{\mathrm{lc},\mathrm{c}}$ (Every locally countable connected graph has a maximal independent set). $\circ$ CAC${}_1^{{\aleph }_{\alpha }}$ (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...

In set theory without the axiom of choice (AC), we observe new relations of the following statements with weak choice principles. $\circ$ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable. $\circ$ If in a partially ordered set, all chains are finite and all antichains have size ${\aleph }_{\alpha }$, then the set has size ${\aleph }_{\alpha }$ for any regular ${\aleph }_{\alpha }$. $\circ$ Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. $\circ$ Every partially ordered set...