Hypergraphs with a quarter uniform Turán density

Abstract

The uniform Turán density $\ddot{I}{€}_1(F)$ 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 $\ddot{I}{€}_1(F)$ for given hypergraphs $F$ was suggested by Erdős and Sós in 1980s. In particular, they raised the questions of determining $\ddot{I}{€}_1(K_4^{(3)-})$ and $\ddot{I}{€}_1(K_4^{(3)})$. 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 Turán density has been determined. In this paper, we give a sufficient condition for $3$-uniform hypergraphs $F$ satisfying $\ddot{I}{€}_1(F)=1/4$. In particular, currently all known $3$-uniform hypergraphs whose uniform Turán density is $1/4$, such as $K_4^{(3)-}$ and the $3$-uniform hypergraphs $F_5^{\star }$ studied in [arXiv:2211.12747], satisfy this condition. Moreover, we find some intriguing $3$-uniform hypergraphs whose uniform Turán density is also $1/4$.

23 pages