Hereditary compactness and discrete subspaces of cardinality $\mathfrak{c}$ If a compact space $K$ is hereditarily separable then it doesn't contain discrete subspaces of size continuum. But, the reverse implication is true? If not, are there easy counterexamples?
Thanks in advance.
 A: Note that we even know that no uncountable discrete subspace exists, which is a stronger conclusion, in general. And the reverse implication in that case is consistently true:
Theorem: if Martin's axiom holds and the Continuum Hypothesis (CH) fails, then a compact Hausdorff space that does not contain an uncountable discrete subspace, is hereditarily separable. 
If the Continuum Hypothesis holds, there exists examples of compact Hausdorff spaces that are hereditarily Lindelöf (so in particular no uncountable discrete subspace exists) but are not separable. These are so-called compact L-spaces. They also exist when a Suslin line exists, e.g., and under other assumptions as well.
So no easy examples, as in some models of set theory we have counterexamples against the reverse implication. And in other models we have a reverse implication for a stronger condition (assuming no uncountable discrete subspace exists).
But if CH fails, we also have a counterexample (albeit a boring one): $\omega_1 + 1$ is compact and Hausdorff (even hereditarily normal), has no subspaces at all of size continuum (if CH fails!) and is not separable. And as said, under CH we have even stronger counterexamples. So in no model of ZFC the reverse implication holds.
So probably some absolute (without any axioms) example can exist (so a non-separable compact Hausdorff space that has no discrete subspace of size continuum), though I cannot think of one now...
