Mention of open problems in statement of purpose
It is a GREAT idea to list some specific problem.  I got a national scholarship and one reason was I wrote about a specific problem I wanted to work on (even with a couple citations to literature).

A little real meat on the bone content helps differentiate from hordes of people who just say how great they are.  

I would maybe just give yourself a little flexibility to say that you will look at other topics of interest when accepted into school and choosing a lab group.