The Sunflower Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $k\geq1$ and let $\mathcal F$ be a finite family of distinct $k$-element subsets of an arbitrary finite set. Three distinct members $A_1,A_2,A_3$ form a three-petal sunflower if \[ A_1\cap A_2=A_1\cap A_3=A_2\cap A_3. \] The common intersection is the core; the sets obtained by removing it are pairwise disjoint.

Statement recorded in the supplied source.

Does there exist an absolute constant $c>0$ such that, for every $k\geq1$ and every such family $\mathcal F$, the inequality \[ |\mathcal F|\geq c^k k! \] forces a three-petal sunflower? [S2] The factorial-bearing assertion above has an affirmative classical answer: the Erdős--Rado sunflower lemma gives the bound $k!2^k$ for three petals, so $c=3$ certainly suffices. The stronger exponential sunflower conjecture removes the factor $k!$; that different assertion is not the recorded target here. [S2]

Short English statement

Does a factorial-times-exponential number of distinct sets of the same size force three sets with identical pairwise intersections?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1235 (COMB-012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ryan Alweiss, Shachar Lovett, Kewen Wu and Jiapeng Zhang, Improved bounds for the sunflower lemma, arXiv:1908.08483v3 (2021), Definition 1.1, Lemma 1.2 and Conjecture 1.3. https://arxiv.org/pdf/1908.08483.