*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?

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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''

schemerather than a single sentence), look at Anton Freund's recentA mathematical commitment without computational strength. $\endgroup$ – Ulrik Buchholtz May 6 at 20:29