TITLE: Natural $\Pi_1$ sentence independent of PA
QUESTION [7 upvotes]: Order invariant graphs and finite incompleteness by Harvey Friedman gives an example of a combinatorial/non-metamathematical $\Pi_1$ sentence that is independent of ZFC. Is there a simpler example of a combinatorial/non-metamathematical $\Pi_1$ sentence that is independent of PA?

REPLY [3 votes]: You may look at Shelah's paper
 ``On logical sentences in PA''. 
For a modern exposition of Shelah's work and an alternative example see ``Independence in Arithmetic: The Method of (L, n)-Models''