Abstract :

Recently, using spectral techniques, H. Huang proved that every subgraph of the hypercube of dimension $n$ induced on more than half the vertices has maximum degree at least $\sqrt(n)$ (square root of $n$). Combined with some earlier work, this completed a proof of the sensitivity conjecture. In this work, we show how to derive Huang's result using only linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of the result. In particular, we prove that in any induced subgraph of the hypercube on more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least $n$ vertices at distance at most 2. As an application we show that for any Boolean function $f$, the polynomial degree of $f$ is bounded above by the product of the $0-$ and $1-$ sensitivities, a strictly stronger statement which implies the sensitivity conjecture.