# 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} }

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

### A Lower Bound for the Complexity of Monotone Graph Properties

- Mathematics, Computer ScienceSIAM 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.

### Evasiveness through a circuit lens

- MathematicsITCS '13
- 2013

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.

### Monotone Properties of k-Uniform Hypergraphs are Weakly Evasive

- MathematicsITCS
- 2015

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.

### Monotone Properties of k-Uniform Hypergraphs Are Weakly Evasive

- MathematicsACM Trans. Comput. Theory
- 2019

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.

### Any monotone property of 3-uniform hypergraphs is weakly evasive

- MathematicsTheor. Comput. Sci.
- 2013

### Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

- Mathematics, Computer ScienceMFCS
- 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.

### On the Sensitivity Complexity of k-Uniform Hypergraph Properties

- Mathematics, Computer ScienceSTACS
- 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).

### On the Power of Parity Queries in Boolean Decision Trees

- Computer ScienceTAMC
- 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.

### Evasive properties of sparse graphs and some linear equations in primes

- MathematicsTheor. Comput. Sci.
- 2014

### List of my favorite publications

- 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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