Minimum spanning tree formulation as integer program The minimum spanning tree problem can be solved in polynomial time via Kruskal's or Prim's algorithm. However, every integer program I have seen that corresponds to the MST problem require a constraint for every subset of vertices, so the number of constraints would be exponential.

That makes me wonder, is it is possible to write down the constraints for a IP corresponding to MST such that the number of constraints is polynomial in the number of vertices? Furthermore, if it is possible, does it hold that every problem in P can be written as a LP with at most polynomial number of constraints?    

Thank you very much!
Yes, it's possible.

It's trivially possible that every problem in P can be reduced to LP in polynomial time, for uninteresting reasons: given an instance of the decision problem, the reduction solves the decision problem (in polynomial time), then either outputs a feasible LP instance or an infeasible LP instance (where these instances are hardcoded).  I realize that's pretty uninteresting and probably not what you wanted.