$C_{10}$ has positive Turán density in the hypercube

Alexandr Grebennikov, João Pedro Marciano

Published 2024-02-29, updated 2024-03-13Version 2

The $n$-dimensional hypercube $Q_n$ 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 $\text{ex}(Q_n,H)$ to be the maximum number of edges of a subgraph of $Q_n$ without a copy of $H$. In this short note, we prove that for any $n \in \mathbb{N}$ $$\text{ex}(Q_n, C_{10}) > 0.024 \cdot e(Q_n).$$ 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.