In the mathematical fields of set theory and extremal combinatorics, a sunflower or -system[1] is a collection of sets in which all possible distinct pairs of sets share the same intersection. This common intersection is called the kernel of the sunflower.
The naming arises from a visual similarity to the botanical sunflower, arising when a Venn diagram of a sunflower set is arranged in an intuitive way. Suppose the shared elements of a sunflower set are clumped together at the centre of the diagram, and the nonshared elements are distributed in a circular pattern around the shared elements. Then when the Venn diagram is completed, the lobe-shaped subsets, which encircle the common elements and one or more unique elements, take on the appearance of the petals of a flower.
The main research question arising in relation to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets) in a given collection of sets? The -lemma, sunflower lemma, and the Erdős-Rado sunflower conjecture give successively weaker conditions which would imply the existence of a large sunflower in a given collection, with the latter being one of the most famous open problems of extremal combinatorics.[2]
Formal definition
Suppose is a set system over , that is, a collection of subsets of a set . The collection is a sunflower (or -system) if there is a subset of such that for each distinct and in , we have . In other words, a set system or collection of sets is a sunflower if all sets in share the same common subset of elements. An element in is either found in the common subset or else appears in at most one of the elements. No element of is shared by just some of the subset, but not others. Note that this intersection, , may be empty; a collection of pairwise disjoint subsets is also a sunflower. Similarly, a collection of sets each containing the same elements is also trivially a sunflower.
Sunflower lemma and conjecture
The study of sunflowers generally focuses on when set systems contain sunflowers, in particular, when a set system is sufficiently large to necessarily contain a sunflower.
Specifically, researchers analyze the function for nonnegative integers , which is defined to be the smallest nonnegative integer such that, for any set system such that every set has cardinality at most , if has more than sets, then contains a sunflower of sets. Though it is not obvious that such an must exist, a basic and simple result of Erdős and Rado, the Delta System Theorem, indicates that it does.
Erdos-Rado Delta System Theorem(corollary of the Sunflower lemma):
For each , , there is an integer such that if a set system of -sets is of cardinality greater than , then contains a sunflower of size .
In the literature, is often assumed to be a set rather than a collection, so any set can appear in at most once. By adding dummy elements, it suffices to only consider set systems such that every set in has cardinality , so often the sunflower lemma is equivalently phrased as holding for "-uniform" set systems.[3]
Sunflower lemma
Erdős & Rado (1960, p. 86) proved the sunflower lemma, which states that[4]
That is, if and are positive integers, then a set system of cardinality greater than or equal to of sets of cardinality contains a sunflower with at least sets.
The Erdős-Rado sunflower lemma can be proved directly through induction. First, , since the set system must be a collection of distinct sets of size one, and so of these sets make a sunflower. In the general case, suppose has no sunflower with sets. Then consider to be a maximal collection of pairwise disjoint sets (that is, is the empty set unless , and every set in intersects with some ). Because we assumed that had no sunflower of size , and a collection of pairwise disjoint sets is a sunflower, .
Let . Since each has cardinality , the cardinality of is bounded by . Define for some to be
Then is a set system, like , except that every element of has elements. Furthermore, every sunflower of corresponds to a sunflower of , simply by adding back to every set. This means that, by our assumption that has no sunflower of size , the size of must be bounded by .
Since every set intersects with one of the 's, it intersects with , and so it corresponds to at least one of the sets in a :
Hence, if , then contains an set sunflower of size sets. Hence, and the theorem follows.[2]
Erdős-Rado sunflower conjecture
The sunflower conjecture is one of several variations of the conjecture of Erdős & Rado (1960, p. 86) that for each , for some constant depending only on . The conjecture remains wide open even for fixed low values of ; for example ; it is not known whether for some .[5] A 2021 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving that for .[6][7] A month after the release of the first version of their paper, Rao sharpened the bound to ;[8] the current best-known bound is .[9]
Sunflower lower bounds
Erdős and Rado proved the following lower bound on . It is equal to the statement that the original sunflower lemma is optimal in .
Theorem.
Proof. For a set of sequence of distinct elements is not a sunflower. Let denote the size of the largest set of -sets with no sunflower. Let be such a set. Take an additional set of elements and add one element to each set in one of disjoint copies of . Take the union of the disjoint copies with the elements added and denote this set . The copies of with an element added form an partition of . We have that,. is sunflower free since any selection of sets if in one of the disjoint partitions is sunflower free by assumption of H being sunflower free. Otherwise, if sets are selected from across multiple sets of the partition, then two must be selected from one partition since there are only partitions. This implies that at least two sets and not all the sets will have an element in common. Hence this is not a sunflower of sets.
A stronger result is the following theorem:
Theorem.
Proof. Let and be two sunflower free families. For each set in F, append every set in to to produce many sets. Denote this family of sets . Take the union of over all in . This produces a family of sets which is sunflower free.
The best existing lower bound for the Erdos-Rado Sunflower problem for is , due to Abott, Hansen, and Sauer.[10][11] This bound has not been improved in over 50 years.
Applications of the sunflower lemma
The sunflower lemma has numerous applications in theoretical computer science. For example, in 1986, Razborov used the sunflower lemma to prove that the Clique language required (superpolynomial) size monotone circuits, a breakthrough result in circuit complexity theory at the time. Håstad, Jukna, and Pudlák used it to prove lower bounds on depth- circuits. It has also been applied in the parameterized complexity of the hitting set problem, to design fixed-parameter tractable algorithms for finding small sets of elements that contain at least one element from a given family of sets.[12]
Analogue for infinite collections of sets
A version of the -lemma which is essentially equivalent to the Erdős-Rado -system theorem states that a countable collection of k-sets contains a countably infinite sunflower or -system.
The -lemma states that every uncountable collection of finite sets contains an uncountable -system.
The -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 is an -sized collection of countable subsets of , and if the continuum hypothesis holds, then there is an -sized -subsystem. Let enumerate . For , let . By Fodor's lemma, fix stationary in such that is constantly equal to on . Build of cardinality such that whenever are in then . Using the continuum hypothesis, there are only -many countable subsets of , so by further thinning we may stabilize the kernel.
See also
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, S2CID 201314765
- Bell, Tolson; Chueluecha, Suchakree; Warnke, Lutz (2021), "Note on Sunflowers", Discrete Mathematics, 344 (7): 112367, arXiv:2009.09327, doi:10.1016/j.disc.2021.112367, MR 4240687, S2CID 221818818
- 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, S2CID 14043028
- 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
- Flum, Jörg; Grohe, Martin (2006), "A Kernelization of Hitting Set", Parameterized Complexity Theory, EATCS Ser. Texts in Theoretical Computer Science, Springer, pp. 210–212, doi:10.1007/3-540-29953-X, ISBN 978-3-540-29952-3, MR 2238686
- Jech, Thomas (2003), Set Theory, Springer
- Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, North-Holland, ISBN 978-0-444-85401-8
- Rao, Anup (2020-02-25), "Coding for Sunflowers", Discrete Analysis, 2020 (2): 11887, doi:10.19086/da.11887, S2CID 202558957
- Rao, Anup (2023), "Sunflowers: from soil to oil" (PDF), Bull. Amer. Math. Soc., 60 (1): 29–38, doi:10.1090/bull/1777
- Shanin, N. A. (1946), "A theorem from the general theory of sets", C. R. (Doklady) Acad. Sci. URSS, New Series, 53: 399–400
- Tao, Terence (2020), The sunflower lemma via Shannon entropy, What's new (personal blog)
External links
- Thiemann, René. The Sunflower Lemma of Erdős and Rado (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)
Notes
- ^ The original term for this concept was "-system". More recently the term "sunflower", possibly introduced by Deza & Frankl (1981), has been gradually replacing it.
- ^ a b "Extremal Combinatorics III: Some Basic Theorems". Combinatorics and more. 28 September 2008. Retrieved 2021-12-10.
- ^ Alweiss et al. (2020), p. 3.
- ^ Kostochka, Alexandr V. (2000), Althöfer, Ingo; Cai, Ning; Dueck, Gunter; Khachatrian, Levon (eds.), "Extremal Problems on Δ-Systems", Numbers, Information and Complexity, Boston, MA: Springer US, pp. 143–150, doi:10.1007/978-1-4757-6048-4_14, ISBN 978-1-4757-6048-4, retrieved 2022-05-02
- ^ Abbott, H.L; Hanson, D; Sauer, N (May 1972). "Intersection theorems for systems of sets". Journal of Combinatorial Theory, Series A. 12 (3): 381–389. doi:10.1016/0097-3165(72)90103-3.
- ^ Alweiss et al. (2020).
- ^ "Quanta Magazine - Illuminating Science". Quanta Magazine. Retrieved 2019-11-10.
- ^ Rao (2020).
- ^ Bell, Chueluecha & Warnke (2021).
- ^ Abbott, H.L; Hanson, D; Sauer, N (May 1972). "Intersection theorems for systems of sets". Journal of Combinatorial Theory, Series A. 12 (3): 381–389. doi:10.1016/0097-3165(72)90103-3.
- ^ Lower Bounds for the Sunflower Problem https://mathoverflow.net/a/420288/12176
- ^ Flum & Grohe (2006).