Minimal bad sequences are necessary for a uniform Kruskal theorem

The minimal bad sequence argument due to Nash-Williams is a powerful tool in combinatorics with important implications for theoretical computer science. In particular, it yields a very elegant proof of Kruskal's theorem. At the same time, it is known that Kruskal's theorem does not require the full strength of the minimal bad sequence argument. This claim can be made precise in the framework of reverse mathematics, where the existence of minimal bad sequences is equivalent to a principle known as $Π^1_1$-comprehension, which is much stronger than Kruskal's theorem. In the present paper we give a uniform version of Kruskal's theorem by relativizing it to certain transformations of well partial orders. We show that $Π^1_1$-comprehension is equivalent to our uniform Kruskal theorem (over $\mathbf{RCA}_0$ together with the chain-antichain principle). This means that any proof of the uniform Kruskal theorem must entail the existence of minimal bad sequences. As a by-product of our investigation, we obtain uniform proofs of several Kruskal-type independence results.

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