Intervals of hypergraph Turán densities
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For each r≥3, does the set of Turán densities of possibly infinite forbidden r-uniform hypergraph families contain an interval of positive length? The stronger target is a terminal interval [1−δᵣ,1].
Why it matters and what counts as progress
Why it matters: An interval would show a continuous range of extremal densities where even positive Hausdorff dimension was previously unknown.
What counts as progress: A construction realizing every density in one positive-length interval, with recursive and lifting steps proved and the allowance of infinite forbidden families retained.
How the problem developed
- 2007 · Open question [1]
Peter Frankl, Vojtěch Rödl, Jie Peng and William Talbot Asked whether the set of Turán densities of r-uniform hypergraph families contains intervals for r≥3.
Codruţ Grosu Asked whether the Hausdorff dimension of the set of Turán densities of possibly infinite r-graph families is positive.
- 25 May 2026 · Theorem [1]
Xizhi Liu and Oleg Pikhurko Proved that for every r≥3, the set of Turán densities of possibly infinite r-graph families contains nondegenerate intervals, including a terminal interval below 1; this also gives Hausdorff dimension 1.
AI contribution
Intervals of hypergraph Turán densities
Liu and Pikhurko use a Ramanujan-graph spectral 3-graph gadget to build a smoothly decreasing sequence of Lagrangians. With their separate recursive and lifting constructions, they show that for every r≥3, Turán densities of possibly infinite forbidden r-graph families contain a terminal interval [1−δ_r,1].
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