Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

Banerjee, Amitayu (2023) Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC. COMMENTATIONES MATHEMATICAE UNIVERSITATIS CAROLINAE. ISSN 0010-2628

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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. • Plf,c (Every locally finite connected graph has a maximal independent set). • Plc,c (Every locally countable connected graph has a maximal independent set). • CACאα (If in a partially ordered set all antichains are finite and all chains have size אα, then the set has size אα) if אα is regular. • CWF (Every partially ordered set has a cofinal well-founded subset). • If G = (VG, EG) is a connected locally finite chordal graph, then there is an ordering <of VG such that {w < v : {w, v} ∈ EG} is a clique for each v ∈ VG.

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