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.
Keywords: chromatic number of product of graphs; ultrafilter lemma; permutation model; Dilworth's theorem; chain; antichain; Loeb's theorem; application of Loeb's theorem
DOI: DOI 10.14712/1213-7243.2021.028
AMS Subject Classification: 05C15 03E25 03E35