Beyond the broken tetrahedron

Here we consider the hypergraph Tur\'an problem in uniformly dense hypergraphs as was suggested by Erd\H{o}s and S\'os. Given a $3$-graph $F$, the uniform Tur\'an density $\pi_u(F)$ of $F$ is defined as the supremum over all $d\in[0,1]$ for which there is an $F$-free uniformly $d$-dense $3$-graph, where uniformly $d$-dense means that every linearly sized subhypergraph has density at least $d$. Recently, Glebov, Kr\'al', and Volec and, independently, Reiher, R\"odl, and Schacht proved that $\pi_u(K_4^{(3)-})=\frac{1}{4}$, solving a conjecture by Erd\H{o}s and S\'os. Despite substantial attention, the uniform Tur\'an density is still only known for very few hypergraphs. In particular, the problem due to Erd\H{o}s and S\'os to determine $\pi_u(K_4^{(3)})$ remains wide open. In this work, we determine the uniform Tur\'an density of the $3$-graph on five vertices that is obtained from $K_4^{(3)-}$ by adding an additional vertex whose link forms a matching on the vertices of $K_4^{(3)-}$. Further, we point to two natural intermediate problems on the way to determining $\pi_u(K_4^{(3)})$, and solve the first of these.