Evasiveness and the Distribution of Prime Numbers

Abstract. A Boolean function on N variables is called evasive if its decision-tree complexity is N. A sequence Bn of Boolean functions is eventually evasive if Bn is evasive for all sufficiently large n. We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla’s conjecture on Dirichlet primes implies that (a) for any graph H, “forbidden subgraph H” is eventually evasive and (b) all nontrivial monotone properties of graphs with ≤ n3/2−ǫ edges are eventually evasive. (n is the number of vertices.) While Chowla’s conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet’s L functions), we show (b) with the bound O(n5/4−ǫ ) under ERH. We also prove unconditional results: (a′ ) for any graph H, the query complexity of “forbidden subgraph H” is `n 2 ´ −O(1); (b′ ) for some constant c > 0, all n...

• Type: Conference
• Year: 2010
• Where: STACS
• Authors: László Babai, Anandam Banerjee, Raghav Kulkarni, Vipul Naik
• Keywords: Chowla’s Conjecture, Forbidden Subgraph H, Nontrivial Monotone Properties, STACS 2010, Theoretical Computer Science