TITLE: A strictly decreasing function between uncountable subsets of the reals
QUESTION [8 upvotes]: By a standard technique of inductive killing everything relevant (in this case decreasing homeomorphisms between uncountable $G_\delta$-subsets of the real line) it is possible to prove the following fact.

Theorem (CH). Under CH the real line contains an uncountable subset $X$ admitting no strictly decreasing function $f:Z\to X$, defined on some uncountable subset $Z$ of $X$.

On the other hand, a known PFA-results of Baumgartner (about the order isomorphness of any $\aleph_1$-dense subsets of the real line) implies the following

Theorem (PFA). Under PFA, for any uncountable subset $X\subset\mathbb R$ there exists a strictly decreasing function $f:Z\to X$, defined on some uncountable subset $Z\subset X$. 

Now

Question. Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)?

REPLY [3 votes]: In Todorcevic's book "Partition Problems in Topology" I have found Proposition 8.4(c) saying that under OCA for any uncountable sets $X,Y$ of reals there exists a strictly increasing function $f:Z\to Y$ defined on some uncountable subset $Z$ of $X$.
This proposition implies that for any uncountable set $X\subset \mathbb R$ there exists a strictly increasing function $f:Z\to -X$ defined on some uncountable subset $Z\subset X$. Then the function $-f:Z\to X$ is stricly decreasing. 

Therefore the PFA-theorem in OP holds under OCA.