Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture

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 root n. This is the best possible result, and it improves a logarithmic lower bound shown by Chung, Furedi, 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.