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.



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> **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 $X$.
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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



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> **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 $Z\subset X$. 
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Now



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> **Question.** Can this PFA-theorem be proved under a weaker assumption like OCA or (MA$+\neg$ CH)? 
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A strictly decreasing function between uncountable subsets of the reals
In Todorcevic's book ["Partition Problems in Topology"](https://books.google.com.ua/books/about/Partition_Problems_in_Topology.html?id=k6UZnZ_qkgIC&redir_esc=y) 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. 



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> Therefore the PFA-theorem in OP holds under OCA. 
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