Sunflower (mathematics)

In mathematics, a sunflower or ${\displaystyle \Delta }$-system[1] is a collection of sets whose pairwise intersection is constant. This constant intersection is called the kernel of the sunflower.

For other uses, see sunflower (disambiguation).

A mathematical sunflower can be pictured as a flower. The kernel of the sunflower is the brown part in the middle, and each set of the sunflower is the union of a petal and the kernel.

The main research question related to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets)? The ${\displaystyle \Delta }$-lemma, sunflower lemma, and sunflower conjecture give various conditions that imply the existence of a large sunflower in a given collection of sets.

Formal definition

Suppose ${\displaystyle W}$ is a set system, that is, a collection of subsets of a set ${\displaystyle U}$. The collection ${\displaystyle W}$ is a sunflower (or ${\displaystyle \Delta }$-system) if there is a subset ${\displaystyle S}$ of ${\displaystyle U}$ such that for each distinct ${\displaystyle A}$ and ${\displaystyle B}$ in ${\displaystyle W}$, we have ${\displaystyle A\cap B=S}$. In other words, ${\displaystyle W}$ is a sunflower if the pairwise intersection of each set in ${\displaystyle W}$ is constant. Note that this intersection, ${\displaystyle S}$, may be empty; a collection of disjoint subsets is also a sunflower.

Sunflower lemma and conjecture

Erdős & Rado (1960, p. 86) proved the sunflower lemma, stating that if ${\displaystyle a}$ and ${\displaystyle b}$ are positive integers then a collection of ${\displaystyle b!a^{b+1}}$ sets of cardinality at most ${\displaystyle b}$ contains a sunflower with more than ${\displaystyle a}$ sets.

The sunflower conjecture is one of several variations of the conjecture of Erdős & Rado (1960, p. 86) that the factor of ${\displaystyle b!}$ can be replaced by ${\displaystyle C^{b}}$ for some constant ${\displaystyle C}$. A 2020 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving the result for ${\displaystyle C=(\log b)^{1+o(1)}}$ (Alweiss et al. 2020).[2]

Analogue for infinite collections of sets

The ${\displaystyle \Delta }$-lemma states that every uncountable collection of finite sets contains an uncountable ${\displaystyle \Delta }$-system.

The ${\displaystyle \Delta }$-lemma is a combinatorial set-theoretic tool used in proofs to impose an upper bound on the size of a collection of pairwise incompatible elements in a forcing poset. It may for example be used as one of the ingredients in a proof showing that it is consistent with Zermelo–Fraenkel set theory that the continuum hypothesis does not hold. It was introduced by Shanin (1946).

If ${\displaystyle W}$ is an ${\displaystyle \omega _{2}}$-sized collection of countable subsets of ${\displaystyle \omega _{2}}$, and if the continuum hypothesis holds, then there is an ${\displaystyle \omega _{2}}$-sized ${\displaystyle \Delta }$-subsystem. Let ${\displaystyle \langle A_{\alpha }:\alpha <\omega _{2}\rangle }$ enumerate ${\displaystyle W}$. For ${\displaystyle \operatorname {cf} (\alpha )=\omega _{1}}$, let ${\displaystyle f(\alpha )=\sup(A_{\alpha }\cap \alpha )}$. By Fodor's lemma, fix ${\displaystyle S}$ stationary in ${\displaystyle \omega _{2}}$ such that ${\displaystyle f}$ is constantly equal to ${\displaystyle \beta }$ on ${\displaystyle S}$. Build ${\displaystyle S'\subseteq S}$ of cardinality ${\displaystyle \omega _{2}}$ such that whenever ${\displaystyle i<j}$ are in ${\displaystyle S'}$ then ${\displaystyle A_{i}\subseteq j}$. Using the continuum hypothesis, there are only ${\displaystyle \omega _{1}}$-many countable subsets of ${\displaystyle \beta }$, so by further thinning we may stabilize the kernel.

See also

• Cap set

References

• Alweiss, Ryan; Lovett, Shachar; Wu, Kewen; Zhang, Jiapeng (June 2020), "Improved bounds for the sunflower lemma", Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, Association for Computing Machinery, pp. 624–630, arXiv:1908.08483, doi:10.1145/3357713.3384234, ISBN 978-1-4503-6979-4
• Deza, M.; Frankl, P. (1981), "Every large set of equidistant (0,+1,–1)-vectors forms a sunflower", Combinatorica, 1 (3): 225–231, doi:10.1007/BF02579328, ISSN 0209-9683, MR 0637827
• Erdős, Paul; Rado, R. (1960), "Intersection theorems for systems of sets", Journal of the London Mathematical Society, Second Series, 35 (1): 85–90, doi:10.1112/jlms/s1-35.1.85, ISSN 0024-6107, MR 0111692
• Jech, Thomas (2003), Set Theory, Springer
• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 978-0-444-85401-8
• Shanin, N. A. (1946), "A theorem from the general theory of sets", C. R. (Doklady) Acad. Sci. URSS (N.S.), 53: 399–400
• Tao, Terence (2020), The sunflower lemma via Shannon entropy, What's new (personal blog)

Notes

1. The original term for this concept was "${\displaystyle \Delta }$-system". More recently the term "sunflower", possibly introduced by Deza & Frankl (1981), has been gradually replacing it.
2. "Quanta Magazine - Illuminating Science". Quanta Magazine. Retrieved 2019-11-10.