Any monotone property of 3-uniform hypergraphs is weakly evasive

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 } ( n 2 ) ? { 0 , 1 } of n-vertex graphs has D ( P ) = ? ( n 2 ) .We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P : { 0 , 1 } ( n 3 ) ? { 0 , 1 } of n-vertex 3-uniform hypergraphs has D ( P ) = ? ( n 3 ) .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.