What mathematics cannot be reduced to pigeonhole?

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Pigeonhole is a fundamental principle without which state of mathematics will be much different. However what examples of good mathematics has not yet been proved and cannot be proved with pigeonhole alone?

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The infinite pigeonhole principle is equivalent to B$\Sigma^0_2$ in the sense of reverse mathematics.

This implies that many theorems such as Heine-Borel, Konig's lemma, Ramsey's theorem are not provable from this Infinite Pigeonhole Principle.