Hypergraphs with zero uniform Tur\'an density at intermediate levels

arXiv:2611.06964v1 Announce Type: new Abstract: An $r$-uniform hypergraph $H$ is uniformly dense with respect to $j$-graphs if, up to a small error, at least a prescribed proportion of the $r$-cliques of every $j$-uniform hypergraph on the same vertex set are edges of $H$. For a forbidden hypergraph $F$, its uniform Tur\'an density at level $j$ is the supremum of these density bounds over arbitrarily large $F$-free hypergraphs with the error tending to zero. For every $r\ge4$ and $1\le j\le r-3$, this paper proves that this Tur\'an density is zero precisely when $F$ admits a vertex ordering in which every $(j+1)$-set shared by two edges occupies the same positions in both edges. This generalizes the zero-density characterizations of Erd\H{o}s at $j=0$ and of Reiher, R\"odl, and Schacht at $j=r-2$, and establishes the latter authors' proposed extension to the remaining nontrivial levels. Consequently, at level $j$, every positive uniform Tur\'an density of a possibly infinite forbidden family is at least $N^{-N}$, where $N=\binom r{j+1}$. The proof combines hypergraph regularity with adjacent exchanges in spreadable random hypergraphs.

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