Any monotone property of 3-uniform hypergraphs is weakly evasive

Abstract

© 2014 Elsevier B.V. For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [11] show that any non-constant monotone property P:{0,1}(n2)→{0,1} of n-vertex graphs has D(P)=Ω(n2).We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P:{0,1}(n3)→{0,1} of n-vertex 3-uniform hypergraphs has D(P)=Ω(n3).Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant [6]. Interestingly, our proof makes use of Vinogradov's Theorem (weak Goldbach Conjecture), inspired by its recent use by Babai et al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question