# Evasiveness and the Distribution of Prime Numbers

@article{Babai2010EvasivenessAT,
title={Evasiveness and the Distribution of Prime Numbers},
author={L{\'a}szl{\'o} Babai and Anandam Banerjee and Raghav Kulkarni and Vipul Naik},
journal={ArXiv},
year={2010},
volume={abs/1001.4829}
}
• Published 26 January 2010
• Mathematics
• ArXiv
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…
13 Citations
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 2013
It is shown that all nontrivial monotone graph properties are evasive, i.e., have decision tree complexity $\binom{n}{2}$ and a lower bound of $\frac{1}{3}n^2-o( n^2)$ for general $n$ is proved.
This paper studies a weakening of the Evasiveness Conjecture called weak-EC, which asserts that every non-trivial monotone transitive Boolean function must have D(f) ≥ n1- ε, for every ε > 0.
Inspired by the outline of the KQS approach, the general framework of "orbit augmentation sequences" of sets with group actions is formalized, showing that a parameter of such sequences is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups involved in the orbit augmentation sequence, assuming all those groups are p-groups.
This work formalizes the general framework of “orbit augmentation sequences” of sets with group actions and shows that a parameter of such sequences, called the spacing, is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups Γ involved in the orbit augmentation sequence, assuming all those groups are p-groups.
• Mathematics, Computer Science
MFCS
• 2016
The answer is no for any hereditary property with {finitely many} forbidden subgraphs by exhibiting a bound of $\Omega(n^{1/k})$ for some constant $k$ depending only on the property.
• Mathematics, Computer Science
STACS
• 2017
This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Ω (nk/2), and shows that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O(N1/3).
• Computer Science
TAMC
• 2015
It is shown that the parity queries can be replaced by ordinary ones at the cost of the total influence aka average sensitivity per query, which is tight as demonstrated by the parity function.
• Mathematics
• 2014
[114] László Babai. Vertex-transitive graphs and vertex-transitive maps. games: A randomized proof system and a hierarchy of complexity classes. On the orders of primitive groups with restricted

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A graph property G is a collection of graphs closed under isomorphism. G is said to be evasive if, for every possible local search strategy, there is at least one graph for which membership in G
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STACS
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