# Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture

## TL;DR

This paper proves a tight bound on the maximum degree of certain subgraphs of hypercubes and uses this to resolve the long-standing Sensitivity Conjecture by establishing a polynomial relation between sensitivity and degree of boolean functions.

## Contribution

It introduces a novel bound on induced subgraphs of hypercubes and applies this to prove the Sensitivity Conjecture, a major open problem in theoretical computer science.

## Key findings

- Maximum degree of certain hypercube subgraphs is at least √n.
- Established polynomial relation between sensitivity and degree of boolean functions.
- Resolved the Sensitivity Conjecture, confirming a key theoretical link.

## 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 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.