C₁₀ has positive Tur\'an density in the hypercube

The n-dimensional hypercube Qₙ is a graph with vertex set \0,1\ⁿ and there is an edge between two vertices if they differ in exactly one coordinate. For any graph H, define ex(Qₙ,H) to be the maximum number of edges of a subgraph of Qₙ without a copy of H. In this short note, we prove that for any n ∈ N ex(Qₙ, C₁₀) > 0.024 · e(Qₙ). Our construction is strongly inspired by the recent breakthrough work of Ellis, Ivan, and Leader, who showed that "daisy" hypergraphs have positive Tur\'an density with an extremely clever and simple linear-algebraic argument.