On the Turán 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 Turán density of $H$ is defined by $π_{\square}(H) = \lim_{n \rightarrow \infty} \operatorname{ex}(Q_n,H)/|E(Q_n)|$. In this short note, we prove \[ π_{\square}(C_{10})\leq π_{\square}(C_6). \] Combined with Baber's upper bound on $π_{\square}(C_6)$, this yields $π_{\square}(C_{10})\leq 0.36577$. 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 $Q_n$.

Disclosure

“≤ . n − 1 e( Qn ) n−1 The last inequality follows since ex( Qn , C10 ) ≤ e( Qn ). This proves Corollary 3.1. Open Question. Is π□ (C10 ) = π□ (C6 )? AI Disclosure. OpenAI’s ChatGPT-5.6 Plus assisted in developing the connection between the C6 -counting argument and the subcube-averaging argument as used in the final part of the proof of Theorem 1.1. That part was subsequently checked, shortened, and reworked by the a”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:12:09.752190Z

Structural counts