Rational Exponents Conjecture

The rational exponents conjecture of Erdős and Simonovits asserts that every rational number occurs as an exponent of the extremal number of a single graph. More precisely, there is a graph for which

where big-Theta notation describes the growth as . Here is the maximum number of edges in an -vertex simple graph containing no copy of . The formulation for is problem 571 in the Erdős problems collection.

Conlon et al. (2026) proved the conjecture for a single forbidden graph, rather than merely a finite family of forbidden graphs. Conlon et al. (2026) present a proof originating in Epoch AI's benchmarking of GPT-6 Astra and explain its dependence on earlier extremal constructions.

Liu and Dai (2026) extended this result to uniform hypergraphs. For every integer and every rational number , there is a single -uniform hypergraph for which . Here is the maximum number of hyperedges in a -uniform hypergraph on vertices containing no copy of . The authors report model assistance in proof discussions and formalization. Their Lean proof assumes two inputs, one collecting the literature and suspension steps, and the other a uniform dichotomy for counts over finite fields.