On the Sensitivity Complexity of k-Uniform Hypergraph Properties

In this paper 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 [1], which conjectures that the sensitivity complexity of kuniform hypergraph properties is at least Ω(nk/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(N1/3), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k.