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 $r\geq3$ and $k\geq1$ be integers. A finite $k$-uniform set family $\mathcal F$ consists of distinct $k$-element subsets of a finite set $X$. An $r$-sunflower consists of distinct $A_1,\ldots,A_r\in\mathcal F$ for which there is a set $C$ satisfying $A_i\cap A_j=C$ for all distinct $i,j$; equivalently, the petals $A_i\setminus C$ are pairwise disjoint.

Statement recorded in the supplied source.

For every integer $r\geq3$, does there exist a constant $c_r>0$, depending only on $r$, such that for every $k\geq1$ and every finite $k$-uniform family $\mathcal F$, \[ |\mathcal F|\geq c_r^k\quad\Longrightarrow\quad\mathcal F\text{ contains an }r\text{-sunflower}? \] The constant is independent of $k$ and of the ground set $X$. [S2]

Short English statement

For a fixed number of petals, does an exponential number of distinct equally sized sets always force a sunflower?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1324 (COMB-006). 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 and Conjecture 1.3. https://arxiv.org/pdf/1908.08483.