An improved hypergraph Mantel's Theorem

In a recent paper, Chao and Yu used an entropy method to show that the Turán density of a certain family $\mathcal{F}$ of $\lfloor r\mathrm{/}2\rfloor$ triangle-like $r$-uniform hypergraphs is $r!\mathrm{/}{r}^r$. Later, Liu determined for large $n$ the exact Turán number $\text{ex}(n,\mathcal{F})$ of this family, and showed that the unique extremal graph is the balanced complete $r$-partite $r$-uniform hypergraph. These two results together can be viewed as a hypergraph version of Mantel’s Theorem. In this paper, building on their methods, we improve both of these results by showing that they still hold with a subfamily ${\mathcal{F}}^{\mathrm{\prime }}\subset \mathcal{F}$ of size $\lceil r\mathrm{/}e\rceil$ in place of $\mathcal{F}$.