Monotonically normal spaces have some very strong properties, despite their modest-seeming definition. This talk has to do with some of these properties, and with set-theoretic consistency and independence results having to do with them. Some of these independence results have to do with monotonically normal spaces themselves. Others are about some classes of more general spaces that have some of the same strong properties, sometimes with additional assumptions. Here is a trio of pairs of theorems illustrating this last theme.

Theorem 1A: Every monotonically normal manifold of dimension >1 is metrizable.
Theorem 1B: If PFA(S)[S], then every hereditarily normal manifold of dimension >1 is metrizable.

Theorem 2A: Every locally compact monotonically normal space is either paracompact or has a closed copy of a regular uncountable cardinal.
Theorem 2B: If PFA + Axiom R, then every locally compact, hereditarily strongly collectionwise Hausdorff space is either paracompact or has a copy of ω1.

Theorem 3A: Every monotonically normal space has a normal product with [0,1].
Theorem 3B: If PFA(S)[S], then every hereditarily normal, locally compact space of countable extent has a normal product with [0,1].

Every monotonically normal space is hereditarily collectionwise normal, so each B theorem features
a significant weakening of monotone normality.