Abstract
We show that the non-existence of mad families is equiconsistent with \(\textit{ZFC}\), answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that \(\textit{ZF}+\textit{DC}+\) “there is no maximal independent set in G” is equiconsistent with \(\textit{ZFC}+\) “there exists an inaccessible cardinal”.
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Partially supported by European Research Council Grant 338821.
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Horowitz, H., Shelah, S. On the non-existence of mad families. Arch. Math. Logic 58, 325–338 (2019). https://doi.org/10.1007/s00153-018-0640-5
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DOI: https://doi.org/10.1007/s00153-018-0640-5