Deterministic communication complexity vs partition number
This question has just been resolved! As I mentioned, it was known that 


  $Pn(f) \leq D(f) \leq (Pn(f))^2$,


but it was a major open problem to show that either $Pn(f) = \Theta(D(f))$ or that there exists a function for which $Pn(f) = o(D(f))$. 

A few days ago this was resolved by Mika Göös, Toniann Pitassi, Thomas Watson (http://eccc.hpi-web.de/report/2015/050/). They show that there exists a function $f$ which satisfies


  $Pn(f) = \tilde{O}((D(f))^{2/3})$.


They also show an optimal result for the one-sided version of $Pn(f)$, which I'll denote by $Pn_1(f)$, where you only need to cover the 1-inputs with rectangles. $Pn_1(f)$ also satisfies  


  $Pn_1(f) \leq D(f) \leq (Pn_1(f))^2$,


and they show that this is the best possible relation between the two measures, since they exhibit a function $f$ which satisfies


  $Pn_1(f) = \tilde{O}((D(f))^{1/2})$.