An improved hypergraph Mantel's Theorem
Daniel Iľkovič, Jun Yan
TL;DR
The paper advances the hypergraph Mantel-type theory by showing that a smaller family of triangle-like tents $\mathcal{F}_{r,k}$ with $k=\lceil r/e\rceil$ already attains the universal Turán density $\pi(\mathcal{F}_{r,k})=\frac{r!}{r^r}$, generalizing Mantel’s triangle result to $r$-uniform hypergraphs. Leveraging the Chao–Yu entropy framework, it reduces the density problem to an optimisation over ratio sequences, with the maximiser uniquely given by $x_i=i/r$, thereby matching the known lower bound. The density bound then yields the exact Turán number for large $n$ via a robust stability framework, establishing that the unique extremal construction is the balanced complete $r$-partite hypergraph $T^r(n)$ for sufficiently large $n$. The work also explores generalised tents and $L$-intersecting hypergraphs, deriving additional corollaries and proposing directions for extending the method to broader subfamilies and configurations. Overall, the paper sharpens the connection between entropy methods and extremal hypergraph theory, providing precise density and extremal results and outlining paths for further generalisation.
Abstract
In a recent paper, Chao and Yu used an entropy method to show that the Turán density of a certain family $\mathcal{F}$ of $\lfloor r/2\rfloor$ triangle-like $r$-uniform hypergraphs is $r!/r^r$. 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 $\mathcal{F}'\subset\mathcal{F}$ of size $\lceil r/e\rceil$ in place of $\mathcal{F}$.