\( \def\<#1>{\left<#1\right>} \newcommand{\R}{\mathcal{R}} \renewcommand{\deg}[2]{\mathrm{deg}_{#2}(#1)} \newcommand{\kw}[1]{\mathfrak{c}(#1)} \)

Sensitivity conjecture (Problem 14.16) solved

Let, as in the book, $Q_n$ be the graph whose vertices are vectors in $\{0,1\}^n$, and two vectors are adjacent if they differ in exactly one coordinate.
Theorem (Huang [1]): Every induced subgraph of $Q_n$ with more than $2^{n-1}$ vertices has a vertex of degree at least $\sqrt{n}$.
By the Gostman-Linial result (Thm. 14.18), this implies the inequality $s(f)\geq \sqrt{\mbox{deg}(f)}$ for every Boolean function $f$, where $s(f)$ is the sensitivity of $f$, and $\mbox{deg}(f)$ is the degree of $f$ [the minimum degree of a multilinear real polynomial $p$ that takes the same values as $f$ on all binary vectors].

The proof in [1] is amazingly simple! See also its one-page version by Knuth [2] or a survey paper [3] or search for "sensitivity" on ECCC for subsequent papers. Or just visit the author's home page for links to blogs on which this result was discussed.


S. Jukna, 27.04.2020