Sunflower (mathematics)

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.

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

The main research question related to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets)? The $\Delta$ -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 $U$ is a universe set and $W$ is a collection of subsets of $U$ . The collection $W$ is a sunflower (or $\Delta$ -system) if there is a subset $S$ of $U$ such that for each distinct $A$ and $B$ in $W$ , we have $A\cap B=S$ . In other words, $W$ is a sunflower if the pairwise intersection of each set in $W$ is constant.

Δ-lemma

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

The $\Delta$ -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).

Δ-lemma for ω₂

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

Sunflower lemma and conjecture

Erdős & Rado (1960, p. 86) proved the sunflower lemma, stating that if $a$ and $b$ are positive integers then a collection of $b!a^{b+1}$ sets of cardinality at most $b$ contains a sunflower with more than $a$ sets. The sunflower conjecture is one of several variations of the conjecture of (Erdős & Rado 1960, p. 86) that the factor of $b!$ can be replaced by $C^{b}$ for some constant $C$ .

For $a=2$ this is now known to be true, with constant $C<2$ .^[2]

References

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 0-444-85401-0.
Shanin, N. A. (1946), "A theorem from the general theory of sets", C. R. (Doklady) Acad. Sci. URSS (N.S.), 53: 399–400

Notes

↑ The original term for this concept was " $\Delta$ -system". More recently the term "sunflower", possibly introduced by Deza & Frankl (1981), has been gradually replacing it.
↑ Kalai, Gil. "The Erdos-Szemeredi Sunflower Conjecture is Now Proven". Retrieved 13 May 2017.

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[1] The original term for this concept was " $\Delta$ -system". More recently the term "sunflower", possibly introduced by Deza & Frankl (1981), has been gradually replacing it.

[2] Kalai, Gil. "The Erdos-Szemeredi Sunflower Conjecture is Now Proven". Retrieved 13 May 2017.