Asymptotically attaining the Moore bound
For positive integers and , let be the maximum order of a graph of maximum degree at most and diameter at most . It is shown thatfor every fixed , thereby resolving the asymptotic degree-diameter problem for fixed diameter.
Also proved a similar lower bound on the edge-variant of the problem, and a tight asymptotic for the bipartite variant of the edge problem.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Combinatorics
- Posed by
- Béla Bollobás
- Year posed
- 1978
- Years open
- 48y
- Solved
- 2026-08-04
- Model
- GPT-5.6
- Vendor
- OpenAI
- Collaborators
- Wouter Cames van Batenburg, Samuel Korsky
- Verification
- Lean-verified
- Publication
- Preprint
- Significance
- 38 / 100
- Disclosed cost
- —
- Wikipedia
- 6 languages
What was actually shown
Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.
What the AI did
The paper's tool disclosure states that GPT-5.6 Pro, used in exploratory brainstorming directed by the authors, suggested splitting complete flags into their odd- and even-rank subflags. That suggestion arose in connection with the edge problem but became the halved-flag construction carrying Theorem 1.1 itself, the graph being named for it. The authors developed it, formulated and verified every argument, and take full responsibility. Generative AI also assisted the Lean 4 formalization.
Verification
Lean 4 formalization at github.com/woutercvb/wewantmoore, checked by the site on 2026-08-06 at commit 32beb227. `DegreeDiameter.theorem_1_1` states Theorem 1.1 itself, as a limit of nKD k d / d^k, and `corollary_1_2` states Corollary 1.2; neither is a weakened lemma, and a second independent route is proved alongside each under `_via_big_cell`. No sorry or admit appears in the sources, and the committed axiom audit shows both final theorems resting only on propext, Classical.choice and Quot.sound. That audit is not taken on trust: the repository's CI builds the project from the pinned toolchain and manifest, regenerates the axiom and dependency reports, and fails if they differ from the committed ones. It passes on this commit. The formalization was itself AI-assisted, per the paper's disclosure, and the authors note it is not a line-by-line transcription: k = 1 is handled by the same construction rather than by complete graphs, and the order and cap asymptotics go through leading terms rather than the displayed O_k(q^-1) estimates.
Sources
Submitted by GoldenMongoose827 on
This honestly seem quite significant. Seems to check out. :)