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