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1 Department of Computer Science 2 Northeastern University Boston

Abstract : 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 $On^{5-4-\epsilon}$ under ERH. We also prove unconditional results: a$-$ for any graph $H$, the query complexity of -forbidden subgraph $H$- is $\binom{n}{2} - O1$; b$-$ for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O1$ 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.

Keywords : decision tree complexity evasiveness graph property group action Dirichlet primes extended Riemann hypothesis

Author: Laszlo Babai - Anandam Banerjee - Raghav Kulkarni - Vipul Naik1 -

Source: https://hal.archives-ouvertes.fr/


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