Since Integer Linear Programming is NP-complete, there is a Karp reduction from any problem in NP to it. I thought this implied that there is always a polynomial-sized ILP formulation for any problem in NP.
But I've seen papers on specific NP problems where people write things like "this is the first poly-sized formulation" or "there is no known poly-sized formulation". That's why I'm puzzled.
This answer is mostly a recap of the comments on the question above.
If a problem is NP-complete, it can indeed be reduced to ILP, by using Karp's reductions (― Joe, andy). Claims of "polynomial sized formulations" from one problem to another, are likely meant as more direct formulations, as opposed to multiple reductions through SAT (― Kaveh).
Yes. Every NP problem has a polynomial-sized ILP formulation.
Here is why. Every NP problem has a polynomial-sized formulation as an instance of SAT. Moreover, all of the usual boolean operators -- logical OR, logical AND, logical NOT, etc. -- can be expressed in ILP, using a constant number of variables and inequalities per boolean operator. See Express boolean logic operations in zero-one integer linear programming (ILP) for details of how to do that. Thus, we get at most a constant-sized blow-up when going from SAT to ILP. This implies that there is a polynomial-sized formulation of every NP problem as an ILP problem.
