Since the continuum has to be a regular/singular uncountable cardinal, and since by Easton Theorem it can be anything we want it to be, what is the forcing that makes the continuum a weakly compact cardinal?
If we can do that then $2^{\aleph_0}$ will have the tree property. But I have to find out first what is the forcing that can make $2^{\aleph_0}$ have the partition property $\kappa \to \(\kappa)^2$ i.e the forcing has to ensure that among the subsets that are added to $2^{\aleph_0}$ we have some $H \subset 2^{\aleph_0}$ of order type $2^{\aleph_0}$ such that $F: $[$2^{\aleph_0}$]$^2 \to 2$ is constant on $[H]^2$ for every such $F$. How do we get such an $H$?