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Abstract
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 has a cofinal well-founded subset -- CWF. $\circ$ Dilworth's decomposition theorem for infinite partially ordered sets of finite width -- DT. We also study a graph homomorphism problem and a problem due to A. Hajnal without AC. Further, we study a few statements restricted to linearly-ordered structures without AC.
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