Hereditary properties of permutations are strongly testable

We show that for every hereditary permutation property P and every ε0 > 0, there exists an integer M such that if a permutation π is ε0far from P in the Kendall’s tau distance, then a random subpermutation of π of order M has the property P with probability at most ε0. This settles an open problem whether hereditary permutation properties are strongly testable, i.e., testable with respect to the Kendall’s tau distance, which is considered to be the edit distance for permutations. Our method also yields a proof of a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio on the relation of the rectangular distance and the Kendall’s tau distance of a permutation from a hereditary property.