GPT-5.4 Pro solved a Ramsey hypergraph problem in FrontierMath for the first time, and also pushed Brian-Larson asymptotics

Of the 15 researcher-level open problems in Epoch AI’s FrontierMath benchmark, the first one has now been solved. GPT-5.4 Pro solved the Ramsey hypergraph partition problem in a free-form conversation, and Epoch AI formally confirmed the result. In a follow-up, the model even went beyond the prompt and established the asymptotic sharpness of the Brian-Larson conjecture.

Claude Opus 4.6 and Gemini 3.1 Pro later solved the same problem inside Epoch’s scaffold, which is a framework that guides the model’s reasoning step by step.

What FrontierMath is

FrontierMath is a benchmark made of open problems or nontrivial known problems written by working mathematicians. Unlike ordinary benchmarks, every task is designed to be code-verifiable, so a model only counts as solving it if it can produce an algorithm that matches the expected answer.

The 15 problems are grouped into four tiers:

Tier	Count	Meaning
Moderately interesting	4	Standard results suitable for a specialist journal
Solid result	5	Good enough for a major journal
Major advance	3	Results with a large impact on the field
Breakthrough	3	Serious problems that have been open for more than 10 years

The topics are six number theory problems, six combinatorics problems, one algebraic geometry problem, and two topology/geometry problems. The Ramsey hypergraph problem that was solved here was in the “Moderately interesting” tier, with a human estimate of one to three months.

The Ramsey hypergraph problem

Ramsey theory studies the idea that large enough structures inevitably contain regular patterns. Here the setting is a hypergraph version of that problem, and the key quantity is a lower-bound improvement for a function called H(n).

In plain terms, a hypergraph is a graph where an edge can connect more than two vertices. A partition is a way to divide a set into non-overlapping groups. In this problem, H(n) asks how large a hypergraph can be if it has no isolated vertices and contains no partition larger than n.

Brian and Larson’s 2019 paper gave a recursive lower bound:

k₁ = 1
kₙ = ⌊n/2⌋ + k_{⌊n/2⌋} + k_{⌈n/2⌉}

They showed that H(n) is at least this large and conjectured that the true growth rate is essentially n·ln n.

The problem statement asked for a constant c > 1 such that H(n) ≥ c·k_n, with the result already holding at n = 15, and it also asked for code returning a witness hypergraph.

GPT-5.4 Pro’s solution

GPT-5.4 Pro found the answer in a free-form conversation with Kevin Barrett and Liam Price. That is important because it was the first time the problem was solved without Epoch’s scaffold.

The core idea was a “frame replacement lemma.” GPT-5.4 Pro introduced the idea of an (n₁, ..., nₜ)-frame, which lets you replace each block with a bounded hypergraph and still keep the whole construction bounded. It acts as a composition rule for recursively building larger witnesses from smaller ones.

The two-direction frame recovers the old Brian-Larson recurrence:

H(a + b) ≥ min(a, b) + H(a) + H(b)

The real improvement came from a four-direction frame based on a 13-edge gadget over four vertices. That extra flexibility creates vertices that contribute to the total size without being consumed by any one partition block, which is what makes the bound beat k_n.

For n = 15, the improvement is already visible:

A₁₅ ≥ A₃ + 3×A₄ + e₃(3) = 5 + 3×8 + 15 = 44
k₁₅ = 43
44/43 > 1

So the requested constant c > 1 exists. The smallest ratio in the paper’s construction is 57/56 at n = 32.

The follow-up generalized the idea to t-direction harmonic families and obtained:

H(n) ≥ ((H_t - 1) / ln t) · n·ln n - O_t(n)

Since (H_t - 1) / ln t → 1, the lower bound approaches:

H(n) ≥ (1 - o(1)) · n·ln n

That matches the Brian-Larson upper bound asymptotically, so the conjectured growth rate is essentially correct.

Model status

After GPT-5.4 Pro solved it first, Epoch AI evaluated several models inside its scaffold.

Model	Variant	Solved
GPT-5.2 Pro	all variants	No
Gemini 3 Deep Think	all problems	No
GPT-5.4 Pro	all problems	Yes, in free-form conversation
GPT-5.4 (xhigh)	all problems	Yes, with scaffold
Claude Opus 4.6 (max)	all problems	Yes, with scaffold
Gemini 3.1 Pro	all problems	Yes, with scaffold

What the mathematician said

Will Brian, one of the problem authors, called the solution exciting and said it strips away inefficiency from the lower-bound construction while reflecting the upper-bound structure quite well. He also said he plans to write the result up as a paper and invite Kevin Barrett and Liam Price as coauthors.

The remaining 14 FrontierMath problems are still open.