3 research outputs found

    Chromatic number of the product of graphs, graph homomorphisms, antichains and cofinal subsets of posets without AC

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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. ◦ If in a partially ordered set, all chains are finite and all antichains are countable, then the set is countable.If in a partially ordered set, all chains are finite and all antichains have size Nα, then the set has size Nα for any regular Nα. Every partially ordered set without a maximal element has two disjoint cofinal sub sets – CS. Every partially ordered set has a cofinal well-founded subset – CWF. 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. © 2021, Commentationes Mathematicae Universitatis Carolinae. All Rights Reserved
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