On the Sensitivity Complexity of $k$-Uniform Hypergraph Properties

TL;DR

This paper explores the sensitivity complexity of hypergraph properties, presenting new bounds that challenge existing conjectures and establishing lower bounds related to the sensitivity conjecture for hypergraph properties.

Contribution

It introduces hypergraph properties with sensitivity complexities that disprove Babai's conjecture and provides lower bounds supporting the sensitivity conjecture for constant $k$.

Findings

Constructed hypergraph properties with sensitivity $O(n^{ ext{ceil}(k/3)})$

Disproved Babai's conjecture on sensitivity lower bounds

Established lower bounds supporting the sensitivity conjecture for constant $k$

Abstract

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a $k$-uniform hypergraph property with sensitivity complexity $O(n^{\lceil k/3\rceil})$ for any $k\geq3$, where $n$ is the number of vertices. Moreover, we can do better when $k\equiv1$ (mod 3) by presenting a $k$-uniform hypergraph property with sensitivity $O(n^{\lceil k/3\rceil-1/2})$. This result disproves a conjecture of Babai~\cite{Babai}, which conjectures that the sensitivity complexity of $k$-uniform hypergraph properties is at least $\Omega(n^{k/2})$. 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. Finally, we give a lower bound for sensitivity of $k$-uniform hypergraph properties, which implies the {\em sensitivity conjecture} of $k$-uniform hypergraph properties for any constant $k$.