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  • $\begingroup$ +1 I suppose it is still possible that a MAD family with no maximal independent set can somehow be constructed in ZFC, without settling the question of completely separable families? There's no reason a MAD family with no maximal independent set would have to be completely separable, is there? By the way, didn't Steve Hechler also do some pioneering work on completely separable families? $\endgroup$ Commented Mar 9, 2021 at 7:56
  • $\begingroup$ Wow - thanks @Will - it's interesting that this digs into an open question! $\endgroup$ Commented Mar 9, 2021 at 8:14
  • $\begingroup$ @bof: You're correct on both counts. I don't see any reason why a MAD family answering Dominic's question needs to be completely separable, and so there might be a relatively easy construction of one in ZFC. And I think Hechler was the first to define completely separable MAD families, even though the question of whether their existence is provable in ZFC is attributed to Erdos & Shelah. $\endgroup$ Commented Mar 9, 2021 at 10:16