On the Tur\'an Density of $C_{10}$ in the Hypercube

Abstract

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 $\operatorname{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Tur\'an density of $H$ is defined by $\pi_{\square}(H) = \lim_{n \rightarrow \infty} \operatorname{ex}(Q_n,H)/|E(Q_n)|$. In this short note, we prove \[ \pi_{\square}(C_{10})\leq \pi_{\square}(C_6). \] Combined with Baber's upper bound on $\pi_{\square}(C_6)$, this yields $\pi_{\square}(C_{10})\leq 0.36577$. 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 $Q_n$.