TITLE: Can a Suslin line be 2-entangled?
QUESTION [15 upvotes]: A Suslin line is a linear order $L$ which is dense with no endpoints, complete, and ccc but not separable.  I'm wondering what kind of order-preserving maps there are from $L$ into $L$.  Specifically,
Question: Can there be a Suslin line $L$ such that for every one-to-one, monotonic function $f$ from an uncountable subset of $L$ into $L$, the set of $x\in\mathrm{dom}(f)$ with $f(x)\neq x$ is countable?
The motivation for this question is the following.  A linear order $L$ is $n$-entangled if for every uncountable set $A$ of pairwise-disjoint $n$-tuples in $L$, and for every $s : n\to 2$, there are $a,b\in A$ such that $a_i < b_i$ if and only if $s(i) = 0$, for all $i < n$.  One can show that a linear order $L$ is $2$-entangled if and only if it is rigid, in the sense that every one-to-one, monotonic function on an uncountable subset of $L$ is equal to the identity on a co-countable subset of its domain.
It's not difficult to show that a weakening of the Open Coloring Axiom implies there are no $2$-entangled sets of reals.  However, OCA is consistent with the existence of a Suslin line.  So an answer to the above question would provide evidence for an answer to the following.
Question: Is OCA consistent with the existence of a $2$-entangled linear order?
Edit: I forgot to mention why entangledness is relevant.  A $3$-entangled linear order is necessarily separable, so $2$-entangledness is the most you might possibly get out of a Suslin line.

REPLY [3 votes]: Regarding the second question:
Definition: $A \subset \mathbb{R}$ of power $\aleph_1$ is called an increasing set if in every uncountable set of pairwise disjoint finite sequences from A there are two sequences $<a_i>$ and $<b_i>$ having the same length such that for all i $a_i<b_i$.
If $A$ is increasing then any strickly order reversing $f:A'\to A$ ($A'$ an uncountable subset of $A$) has a fixpoint.
In their paper "On the consistency of some partition theorems for continuous colorings and the structure of $\aleph_1$-dense real order types" Abraham, Rubin and Shelah prove the consistency of OCA and an increasing set (a somewhat analogue of a 2-entangled set as explained above). Note that Todorcevic' Open Coloring Axiom is a related but differently defined axiom than that of Abraham, Rubin and Shelah.