Hereditary properties of permutations are strongly testable

Abstract

We show that for every hereditary permutation property P and every eps>0, there exists an integer M such that if a permutation p is eps-far from P in the Kendall's tau distance, then a random subpermutation of p of order M has the property P with probability at most eps. This settles an open problem whether hereditary permutation properties are strongly testable, i.e., testable with respect to the Kendall's tau distance. In addition, 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.