Does every NP problem have a poly-sized ILP formulation? 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).