Evasiveness and the Distribution of Prime Numbers

Abstract

A Boolean function on $N$ variables is called \emph{evasive} if its decision-tree complexity is $N$.   A sequence $B_n$ of Boolean functions is \emph{eventually evasive} if $B_n$ 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 $\le n^{3/2-\epsilon}$ 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 $O(n^{5/4-\epsilon})$ under ERH.

We also prove unconditional results: (a$'$) for any graph $H$, the query complexity of ``forbidden subgraph $H$'' is $\binom{n}{2} - O(1)$; (b$'$) for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O(1)$ edges are eventually evasive. 

Even these weaker, unconditional results rely on deep results from number theory such as Vinogradov's theorem on the Goldbach conjecture.  

Our technical contribution consists in connecting the topological  framework of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot, and Shi (2002), with a deeper analysis of the orbital structure of permutation groups and their connection to the distribution of prime numbers. Our unconditional results include stronger versions and generalizations of some result of Chakrabarti et al.