On the Turán Density of $C_{10}$ in the Hypercube

The $n$-dimensional hypercube $Q_n$ is the graph with vertex set $\{0,1{\}}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. For a graph $H$, let $\mathrm{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Turán density of $H$ is defined by ${\pi }_{\mathrm{\square }}(H)={\lim }_{n\to \mathrm{\infty }}\mathrm{ex}(Q_n,H)\mathrm{/}\mathrm{\mid }E(Q_n)\mathrm{\mid }$. In this short note, we prove $${\pi }_{\mathrm{\square }}(C_{10})\le {\pi }_{\mathrm{\square }}(C_6)\mathrm{.}$$ Combined with Baber’s upper bound on ${\pi }_{\mathrm{\square }}(C_6)$, this yields ${\pi }_{\mathrm{\square }}(C_{10})\le 0.36577$. Our proof first bounds the number of copies of $C_6$ in $C_{10}$-free subgraphs of $Q_n$, and then applies an averaging argument over the subcubes of $Q_n$.