[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"w0oxK4tQUW":3},"# FrontierMath Ramsey Hypergraphs in Lean\n\n[Formalizing progress on Epoch AI's FrontierMath open problem on hypergraphs](https://epoch.ai/frontiermath/open-problems/ramsey-hypergraphs).\n\nThis repository formalizes lower bounds for the extremal hypergraph function $H(n)$ introduced by Will Brian and Paul B. Larson in [*Choosing between incompatible ideals*](https://www.sciencedirect.com/science/article/pii/S019566982100041X). The sequence arises as the finitary core of a Ramsey-theoretic problem about incompatible ideals and the simultaneous convergence of infinite series.\n\nA hypergraph $(V,\\mathcal H)$ is said to contain a partition of size $n$ if there are $D \\subseteq V$ and $\\mathcal P \\subseteq \\mathcal H$ with $|D| = n$ such that every member of $D$ belongs to exactly one edge of $\\mathcal P$. The extremal function $H(n)$ is the largest $k$ for which there is a hypergraph with $k$ vertices, no isolated vertices, and no partition of size greater than $n$.\n\nThe FrontierMath page presents warm-up, single-challenge, and full-problem variants. This repository addresses the **full-problem** variant: it formalizes an explicit construction showing\n\n$$\nH(n) \\ge \\frac{26}{25}\\,k_n \\qquad (n \\ge 15),\n$$\n\nwhere Brian–Larson's recursive benchmark is given by\n\n$$\nk_1 = 1,\n\\qquad\nk_n = \\lfloor n/2 \\rfloor + k_{\\lfloor n/2 \\rfloor} + k_{\\lfloor (n+1)/2 \\rfloor}.\n$$\n\nThus the constant-factor improvement is already in effect at $n=15$, exactly as requested in the FrontierMath full problem.\n\n**Asymptotic consequence.** The repository also formalizes GPT-5.4 Pro's Lubell-family lower bound, which implies\n\n$$\n\\liminf_{n\\to\\infty} \\frac{H(n)}{k_n} \\ge 2\\ln 2.\n$$\n\nCombined with Brian–Larson's upper bound (not formalized here)\n\n$$\nH(n) \u003C n\\ln n + \\gamma n + \\tfrac12,\n$$\n\nthis yields the sharp asymptotic formula\n\n$$\n\\lim_{n\\to\\infty} \\frac{H(n)}{n\\ln n} = 1.\n$$\n\n**Note:** on the finite side, the formalized construction gives $H(20) \\ge 65$. So this development resolves the FrontierMath full-problem prompt, while stopping one vertex short of the single-challenge target $H(20) \\ge 66$.\n\nThis Lean development is based on an informal proof produced by **GPT-5.4 Pro**. The prompters of that informal proof were **Kevin Barreto** and **Liam Price**; the original paper output is archived on Google Drive [here](https://drive.google.com/drive/u/1/folders/1CDldqw6_AUrWhFctO-cC21dR2FEXaH9x).\n\n## Highlights\n\n* The substitution theorem underlying the recursive witness constructions is formalized in Lean.\n\n* Explicit uniform lower bound\n\n$$\nH(n) \\ge \\frac{26}{25}\\,k_n\n\\qquad (n \\ge 15).\n$$\n\n* Lubell-frame asymptotic lower bound\n\n$$\nH(n) \\ge \\frac{h_t-1}{\\log_2 t}n\\log_2 n - O_t(n)\n\\qquad \\left(t \\ge 2 \\text{ fixed},\\qquad h_t:=\\sum_{n=1}^{t}\\frac{1}{n}\\right).\n$$\n\n* Asymptotic consequence\n\n$$\n\\liminf_{n\\to\\infty} \\frac{H(n)}{k_n} \\ge 2\\ln 2,\n\\qquad\nk_n = \\frac12 n\\log_2 n + O(n).\n$$\n\n* The repository includes the full paper, blueprint, and the explicit Python constructor required by the FrontierMath problem statement.\n* The current formal development is about 6,300 lines of Lean.\n\n## Repository structure\n\nThe core development is organized as follows:\n\n* [`FrontierMathHypergraphs/Basic.lean`](FrontierMathHypergraphs/Basic.lean) — hypergraph definitions, partitions, the extremal function `H`, and the benchmark sequence `k`.\n* [`FrontierMathHypergraphs/Substitution.lean`](FrontierMathHypergraphs/Substitution.lean) — support patterns, frames, substitution hypergraphs, and the substitution theorem.\n* [`FrontierMathHypergraphs/Uniform.lean`](FrontierMathHypergraphs/Uniform.lean) — the finite bootstrap and the uniform $26/25$ lower bound.\n* [`FrontierMathHypergraphs/Lubell.lean`](FrontierMathHypergraphs/Lubell.lean) — Lubell frames and the asymptotic theory.\n* [`blueprint/src/content.tex`](blueprint/src/content.tex) — blueprint source.\n* [`frontier.tex`](frontier.tex) and [`paper/input.tex`](paper/input.tex) — paper source.\n\n## Key links\n\n* [FrontierMath open problem: *A Ramsey-style Problem on Hypergraphs*](https://epoch.ai/frontiermath/open-problems/ramsey-hypergraphs)\n* [Brian–Larson paper on ScienceDirect](https://www.sciencedirect.com/science/article/pii/S019566982100041X)\n* [Original GPT-5.4 Pro paper output (Google Drive)](https://drive.google.com/drive/u/1/folders/1CDldqw6_AUrWhFctO-cC21dR2FEXaH9x)\n* [Accompanying paper source in this repository](frontier.tex)\n* [Blueprint source in this repository](blueprint/src/content.tex)\n\n## Useful commands\n\nCompile the Lean files (requires [`Lean`](https://docs.lean-lang.org/lean4/doc/quickstart.html)):\n\n```sh\nlake exe cache get && lake build\n```\n\nBuild the blueprint PDF (requires [`uv`](https://docs.astral.sh/uv/)):\n\n```sh\nuvx leanblueprint pdf\n```\n\nBuild and serve the blueprint website:\n\n```sh\nuvx leanblueprint web && uvx leanblueprint serve\n```\n\n## References\n\n* Will Brian and Paul B. Larson, *Choosing between incompatible ideals*, European Journal of Combinatorics **96** (2021), Article 103349. DOI: [10.1016/j.ejc.2021.103349](https://doi.org/10.1016/j.ejc.2021.103349), ScienceDirect: [link](https://www.sciencedirect.com/science/article/pii/S019566982100041X), arXiv: [1908.10914](https://arxiv.org/abs/1908.10914)\n* Epoch AI, *A Ramsey-style Problem on Hypergraphs*. FrontierMath open problem page: [link](https://epoch.ai/frontiermath/open-problems/ramsey-hypergraphs)\n* GPT-5.4 Pro, *A constant-factor lower bound for $H(n)$*. Informal paper output archived on Google Drive; prompters: Kevin Barreto and Liam Price. Archive: [link](https://drive.google.com/drive/u/1/folders/1CDldqw6_AUrWhFctO-cC21dR2FEXaH9x)\n",1780846760266]