Minimal bad sequences are necessary for a uniform Kruskal theorem

• Published in: Advances in mathematics (New York. 1965) 2022-05, Vol.400, p.108265, Article 108265
• Main Authors: Freund, Anton, Rathjen, Michael, Weiermann, Andreas
• Format: Article
• Language: English
• Subjects: Dilators on partial orders; Independence results; Kruskal's theorem; Minimal bad sequence; Recursive path ordering; Reverse mathematics
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2022.108265

Kruskal's theorem on trees is a classical result of combinatorics with important applications in computer science. The minimal bad sequence argument of Nash-Williams yields a proof that is very elegant but not as elementary as possible, by previous results from mathematical logic. In the present paper, we formulate a uniform Kruskal theorem, which extends the original theorem from trees to general recursive data types. Our main result shows that the minimal bad sequence argument does yield the most elementary proof of this uniform Kruskal theorem. So in the uniform case, the elegant proof coincides with the most elementary one. In more precise and technical terms, we work in the framework of reverse mathematics, where we show that the uniform Kruskal theorem is equivalent to Π11-comprehension, over RCA0 extended by the chain-antichain principle. As a by-product of our investigation, we obtain uniform proofs of several Kruskal-type independence results.