THE SUNFLOWER CONJECTURE FALLS

Take the numbers from 1 to n and look at all the ways of picking some of them: there are 2ⁿ such subsets. Now pick a few subsets that share exactly the same common part, and nothing else. Mathematicians call this a sunflower: the shared part is the core, and what remains of each set is a petal that touches no other petal.

Sunflowers are a basic tool of combinatorics. The paper lists links to the limits of certain logic circuits, to the “cap set” problem, to fast matrix multiplication and to information theory.

A question from 1970

The most natural question about them goes back, according to the authors, to a 1970 paper by Paul Erdős. How large can a collection of subsets of {1, …, n} be if no k of its sets form a sunflower?

In the 1970s, Erdős and Endre Szemerédi conjectured that such a collection must be exponentially smaller than the 2ⁿ possible subsets: at most cⁿ, for some number c strictly below 2.

For decades, the best bounds only shaved off a little, never enough to bring the base below 2. Only the three-petal case (k = 3) was settled, in 2016, through the breakthrough on the cap set problem and a reduction due to Alon, Shpilka and Umans.

The bound

Matija Bucić and Fankang He (University of Vienna) and Hyunwoo Lee (KAIST and the Institute for Basic Science, South Korea) prove the conjecture for every k ≥ 3. Their main theorem: a collection with no k-petal sunflower has at most

3 · (2 − 1/(16k³))ⁿ

members. The base sits just below 2, but it is a fixed number below 2, and that is exactly what Erdős and Szemerédi asked for. A construction from 1997 shows that, for large k, the truth cannot be much lower.

From sets to vectors

The proof goes through a version of the problem with vectors. Split the numbers into n blocks of D elements and choose one element per block: each choice is a vector. Several vectors form a sunflower when, in every position, their entries are either all equal or all different.

The authors then build a “sunflower tester”: a function that, on a sunflower-free collection, is non-zero only when all its inputs are the same vector. The new ingredient is to write this function, step by step, with interpolation polynomials of low degree. Counting those polynomials bounds the size of the collection (their Theorem 2). A probabilistic argument, random blocks averaged over all set sizes, carries the bound back to ordinary sets.

Found by a machine, checked by another

The paper’s statement on AI is unusually direct. The proof of Theorem 2 was found by ChatGPT 6 Astra Pro, after a short discussion that began by asking it to generalise the known three-petal argument. The authors needed that step for another goal: an induced version of a graph problem posed by Szemerédi, which they announce for a later note.

The model wrote its proof in the language of “slice rank”. The authors rewrote it in the language of the polynomial method, which they find easier for readers. They state that they verified every result and take full responsibility, and that the proof has been verified in Lean 4.35, a proof-checking software, with the verification to be posted on the Palomar repository.

What it does not settle

The authors did not try to optimise the base. For three petals, it does not beat the 2017 bound of Naslund and Sawin, about 1.89. And the stronger sister problem, the Erdős–Rado sunflower conjecture, remains open: their bounds still depend heavily on the block size D, which is exactly what that problem forbids.