Consider the sentence $\mathtt{PSP}\_\mathfrak{c}$: "Every subset of $\mathbb{R}$ having the cardinality of the continuum contains a Cantor set".  

A priori this sentence is weaker than the usual $\mathtt{PSP}$, since $\mathtt{PSP}\_\mathfrak{c}$ requires the set not only to be uncountable, but to be of the size of the continuum.


My questions are:


* Given a model $M$ of $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}\_\mathfrak{c}$, can we find a forcing notion $\mathbb{P}$ in $M$ such that for any $G$ $\mathbb{P}$-generic over $M$, the extension $M[G]$ satisfies $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}$?
* More in general, can we prove that $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}\_\mathfrak{c}$ is equiconsistent with $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}$?
* Is it consistent $\mathtt{ZF}+\mathtt{DC}+\mathtt{PSP}\_\mathfrak{c}+\neg\mathtt{PSP}$, modulo the consistency of some large cardinal?


Ideas?  

Thanks!



$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$
Here are the answers you're looking for:


1. No.
2. No.
3. Yes, no large cardinals needed! (Which explains the previous two answers.)


Look no further than John Truss' paper:



> 
> *Truss, John*, [**Models of set theory containing many perfect sets**](http://dx.doi.org/10.1016/0003-4843(74)90015-1), Ann. Math. Logic 7, 197-219 (1974). [ZBL0302.02024](https://zbmath.org/?q=an:0302.02024).
> 
> 
> 


There he shows that by just adding many Cohen reals to $L$ we can have a model of $\sf ZF+DC$ in which every set of reals is well-orderable or contains a perfect set. Moreover, we can choose an arbitrarily high Hartogs numbers for the reals, so there can be many well-orderable cardinals intermediate to the continuum.


This provides us with a model of $\sf PSP\_{\frak c}$ that requires no large cardinals. However, since $\sf PSP$ implies that $\omega\_1$ is inaccessible to reals, that means we cannot extend further to a model of full $\sf DC+PSP$ without an inaccessible cardinal present.


