Maximum density of vertex-induced perfect cycles and paths in the hypercube
Let $H$ and $K$ be subsets of the vertex set $V(Q_d)$ of the $d$-cube $Q_d$ (we call $H$ and $K$ configurations in $Q_d$). We say $K$ is an \emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ to $K$. If $d$ is a positive integer and $H$ is a configuration in $Q_d$, we define $\pi(H,d)$ to be the limit as $n$ goes to infinity of the maximum fraction, over all subsets $S$ of $V(Q_n)$, of sub-$d$-cubes of $Q_n$ whose intersection with $S$ is an exact copy of $H$. We determine $\pi(C_8,4)$ and $\pi(P_4,3)$ where $C_8$ is a "perfect" 8-cycle in $Q_4$ and $P_4$ is a "perfect" path with 4 vertices in $Q_3$, and make conjectures about $\pi(C_{2d},d)$ and $\pi(P_{d+1},d)$ for larger values of $d$. In our proofs there are connections with counting the number of sequences with certain properties and with the inducibility of certain small graphs. In particular, we needed to determine the inducibility of two vertex disjoint edges in the family of bipartite graphs.