Abstract: The (binary) $d$-dimensional hypercube $Q^d$ is the graph with vertex set $\{0,1\}^d$ where an edge is drawn between every two vertices/vectors differing in a unique coordinate. We will consider the percolated hypercube $Q^d_p$ where every edge of $Q^d$ is retained independently and with probability $p$. When $pd > 1+\varepsilon$, the existence of a giant component in this model has been known since the early 80's from a work of Ajtai, Komlós and Szemerédi. The talk will mostly discuss a result which takes us one step further: namely, we will explain why, for every $\varepsilon>0$, there exists a constant $C=C(\varepsilon)>0$ such that, if $pd>C$, then $Q^d_p$ typically contains a cycle of length at least $(1-\varepsilon) 2^d$.
Joint work in progress with Michael Anastos, Sahar Diskin, Joshua Erde, Mihyun Kang and Michael Krivelevich.