Rational exponents of bipartite extremal graph numbers

For every rational r with 1≤r<2, is there one finite bipartite graph H with ex(n,H)=Θ(nʳ)? Here ex(n,H) is the maximum edge count of an n-vertex graph with no copy of H.

Why it matters and what counts as progress

Why it matters: Realizing an exponent by one forbidden graph is stronger than realization by a finite forbidden family.

What counts as progress: A single-graph construction for each rational exponent, with matching upper and lower bounds, rather than a result for only selected intervals of r.

How the problem developed

2018 · Partial result [1][3]
Boris Bukh and David Conlon Realized every rational exponent between 1 and 2 using a finite family of forbidden graphs; the single-graph form remained open.
2026 · Reported claim [2][3]
Epoch AI and GPT-6 Astra The repository reports a construction giving each rational r in [1,2) as ex(n,H)=Θ(n^r) for a single finite bipartite graph H.
13 Oct 2026 · Human exposition [3]
Conlon, Janzer, Jiang, Kim, Lee, Liu, Longbrake, and Yepremyan Published a human proof exposition explaining the single-graph rational-exponents result and its rooted-graph construction; it describes the proof claimed in the earlier repository.

AI contribution

GPT-6 Astra proves the rational-exponents conjecture

For every rational r with 1 ≤ r < 2, there is a finite bipartite graph H whose extremal number ex(n,H) grows as Θ(n^r). Here ex(n,H) is the largest number of edges in an n-vertex graph containing no copy of H; Θ means matching upper and lower bounds up to constant factors.

AI involvementAI-autonomous

ImportanceMajor

Technical verificationFormal (Lean)

Human understandingFull