For every r≥3 there exist r-uniform hypergraphs whose uniform Turán density is 1/4, and others whose uniform Turán density is binom(r,2)^{-binom(r,2)}; the r≥5 cases are the first explicit non-zero values.
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Uniform Tur\'an density beyond 3-graphs
For every r≥3 there exist r-uniform hypergraphs whose uniform Turán density is 1/4, and others whose uniform Turán density is binom(r,2)^{-binom(r,2)}; the r≥5 cases are the first explicit non-zero values.