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We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality ℵ1. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that ℤ is a p.i.d. with infinitely many primes is essential for this result.

Hereditarily separable groups and monochromatic uniformization