  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
result is best possible, and improves a logarithmic lower bound shown by Chung,
F\"uredi, 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.
