On high-girth layered graphs of positive Turán density in a hypercube

For a graph $H$, let $\mathrm{ex}(Q_n,H)$ be the largest number of edges in a subgraph of the hypercube $Q_n$ of dimension $n$ that contains no subgraph isomorphic to $H$. The Turán density of $H$ in a hypercube, denoted ${\pi }_{\mathrm{\square }}(H)$, is defined as ${\lim }_{n\to \mathrm{\infty }}\mathrm{ex}(Q_n,H)\mathrm{/}\mathrm{\mid }E(Q_n)\mathrm{\mid }$. Determining ${\pi }_{\mathrm{\square }}(H)$ remains a widely open question for general $H$. Conlon found a large class of graphs with zero Turán density in a hypercube. In this note, we address the case when ${\pi }_{\mathrm{\square }}(H)>0$. If a graph $H$ is not embeddable in an edge-layer of a hypercube, then ${\pi }_{\mathrm{\square }}(H)\ge 1\mathrm{/}2$, as can be seen by taking every other edge layer of $Q_n$. Among the layered graphs, the only ones known to have positive Turán density in a hypercube are graphs containing cycles of length $6$ or $10$. We show that, for every $g\ge 3$, there is a layered graph of girth at least $g$ whose Turán density in a hypercube is at least $1\mathrm{/}2$.