On the Sensitivity Complexity of k -Uniform Hypergraph Properties

Abstract

In this article, we investigate the sensitivity complexity of hypergraph properties. We present a k -uniform hypergraph property with sensitivity complexity O ( n (⌈ k/3 ⌉) for any k ≥ 3 , where n is the number of vertices. Moreover, we can do better when k ≡ 1 (mod 3) by presenting a k -uniform hypergraph property with sensitivity O (n⌈ k/3 ⌉-1/2). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k -uniform hypergraph properties is at least Ω ( n k/2 ). We also investigate the sensitivity complexity of other symmetric functions and show that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O (N 1/3 ), where N is the number of variables.