Turing universality, computability, and incompleteness in hypergraph Tur\'an theory

Abstract

Given a finite family $\mathcal F$ of forbidden $r$-graphs, the Tur\'an problem asks for the maximum asymptotic edge density of $\mathcal F$-free $r$-graphs and the structure of near-extremal examples. We show that both questions can encode arbitrary computation. Fix a universal Turing machine $\mathsf U$. For every sufficiently large fixed $r$, there is a rational $\tau_r\in(0,1)$ such that, from each binary word $\beta$, one can construct a finite family $\mathcal F_{r,\beta}$ with $\pi(\mathcal F_{r,\beta})=\tau_r$ if $\mathsf U$ does not halt on $\beta$, and $\pi(\mathcal F_{r,\beta})>\tau_r$ otherwise. The same dichotomy governs extremal structure. We construct finite families $\mathcal G_{r,\beta}$ such that nonhalting gives a unique extremal limit and Erd\H{o}s--Simonovits stability, whereas halting gives two nonempty compact extremal phases separated by the sign of a fixed continuous statistic. Hence uniqueness and connectedness of the extremal space, symmetry breaking, two-phase behavior, and stability are all undecidable. The reductions are effective and verifiable in ZFC by finite certificates. Consequently, for every consistent computably axiomatized extension of ZFC and every sufficiently large fixed $r$, there is a finite family $\mathcal F$ for which the true equality $\pi(\mathcal F)=\tau_r$ is neither provable nor refutable; analogous independence holds for the five structural properties above. We also obtain effective approximation, classify exact comparison complexity, and show that the smallest improvement witnesses have Busy-Beaver growth, with no uniform computable positive lower bound on the density gain.