$C_{10}$ has positive Turán density in the hypercube
Alexandr Grebennikov, João Pedro Marciano
TL;DR
The paper proves that the 10-cycle has positive Turán density in the hypercube by establishing a concrete lower bound on $ex(Q_n,C_{10})$. It adapts a linear-algebraic random-vector construction, inspired by the daisy-gap method of Ellis–Ivan–Leader, to build large $C_6$-free induced subgraphs within the layer graphs $L_r(n)$ and links these constructions to a bound on $ex^*(Q_n,C_6^-)$, which in turn yields a lower bound on $ex(Q_n,C_{10})$ via a known inequality. The core novelty is a probabilistic, vector-space based scheme that yields $C_6$-free subgraphs with edge density exceeding a positive fraction of $e(L_r(n))$ for odd layers, enabling a global construction with positive density for $C_{10}$. This resolves the long-open case of positive Turán density for $C_{10}$ in the hypercube and contributes to the Ramsey theory of even cycles in $Q_n$ by providing a Ramsey graph with positive Turán density.
Abstract
The $n$-dimensional hypercube $Q_n$ is a graph with vertex set $\{0,1\}^n$ such that there is an edge between two vertices if and only 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 Turán density with an extremely clever and simple linear-algebraic argument.