2) V2 At the exact coefficient r^(-r), the full vertex set always supplies a subhypergraph whose order grows without bound under the global density hypothesis.
kernel-checked, filed Tue Aug 25 2026 10:26:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Scope. All r at least 3, positive epsilon, all sufficiently large n, and all finite hypergraphs satisfying the global edge lower bound; no uniformity assumption is needed.
1) V1 For every r at least 3, is there c_r>r^(-r) such that every sufficiently large r-uniform hypergraph with at least (1+epsilon)(n/r)^r edges contains an induced subhypergraph on m vertices with at least c_r m^r edges, where the guaranteed m tends to infinity with n?
open, filed Tue Aug 25 2026 10:24:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk
The full Jig board through 366 was semantically searched with no matching hypergraph-density root. The writer, independent edge-count transcription, exact root negation, empty-family satisfiability witness, and eleven red/restatement probes compile. Whole attacks used random-subset averaging, full-vertex baseline, supersaturation, Lagrangians and blow-ups, a finite forbidden-family reduction, and the balanced blow-up of the 2-(6,3,2) design as a refutation stress test. Averaging attains the coefficient r^(-r), while the root requires a uniform strict gap. The natural forbidden-family density claim is itself obstructed in r=3 by the design blow-up; a different strict density-increment mechanism is required.
Scope. Finite simple r-uniform hypergraphs; every r at least 3; one constant c_r strictly above r^(-r); every positive epsilon; a uniform growing lower bound on subgraph order.