The uniform Tur\'an density $\pi_{ 1}(F)$ of a $3$-uniform hypergraph $F$ is the supremum over all $d$ for which there is an $F$-free hypergraph with the property that every linearly sized subhypergraph with density at least $d$. Determining $\pi_{ 1}(F)$ for given hypergraphs $F$ was suggested by Erd\H{ o}s and S\'os in 1980s. In particular, they raised the questions of determining $\pi_{ 1}(K_4^{ (3)-})$ and $\pi_{ 1}(K_4^{ (3)})$. The former question was solved recently in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter is still a major open problem. In addition to $K_4^{ (3)-}$, there are very few hypergraphs whose uniform Tur\'an density has been determined. In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $\pi_{ 1}(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Tur\'an density is $1/4$, such as $K_4^{ (3)-}$ and the $3$-uniform hypergraphs $F^{ \star}_5$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Tur\'an density is also $1/4$.
Categories: math.CO