Rational Turán exponents of a single k-uniform hypergraph

For a fixed k-uniform hypergraph F, let exₖ(n,F) denote the maximum number of edges in an F-free k-uniform hypergraph on n vertices. We prove that for every integer k≥2 and every rational r∈[0,1] there exists a k-uniform hypergraph Fk,r such that {equation*} ex_k(n,Fk,r)=Θ(nᵏ⁻ʳ). {equation*} The case $k=2$ is the rational exponents conjecture of Erdős and Simonovits {Erdos1988}, also known as Erdős problem #571 {BloomErdoswebsite}; a proof was announced in September 2026 and is expounded by Conlon et al.\ {Conlon-Janzer-Jiang-Kim-Lee-Liu-Longbrake-Yepremyan}. For k≥3 the theorem extends their result to k-uniform hypergraphs. In the single-hypergraph setting, this strengthens Fitch's finite-family theorem {Fitch} by realizing each exponent with a single forbidden k-uniform hypergraph for 0≤ r≤1. We also show, via an explicit two-edge construction, that for k≥4 this strengthening fails for $ k/2