Abstract
In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt {n}$. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.
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author = {Ambainis, A. and Sun, X.},
title = {New separation between $s(f)$ and $bs(f)$},
journal = {Electronic Colloquium on Computational Complexity (ECCC)},
volume = {18},
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author = {Gilmer, Justin and Kouck\'{y},
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title = {A new approach to the sensitivity conjecture},
booktitle = {I{TCS}'15---{P}roceedings of the 6th {I}nnovations in {T}heoretical {C}omputer {S}cience},
pages = {247--254},
publisher = {ACM, New York},
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title = {Smooth {B}oolean functions are easy: efficient algorithms for low-sensitivity functions},
booktitle = {I{TCS}'16---{P}roceedings of the 2016 {ACM} {C}onference on {I}nnovations in {T}heoretical {C}omputer {S}cience},
pages = {59--70},
publisher = {ACM, New York},
year = {2016},
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}