For any sunflower size, does every set of uniformly sized sets which is of cardinality greater than some exponential in the set size contain a sunflower?
In the mathematical fields of set theory and extremal combinatorics, a sunflower or -system is a collection of sets whose pairwise intersection is constant. This constant intersection is called the kernel of the sunflower.
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.
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 the pairwise intersection of each set in is identical. 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 clear 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:
For each , is an integer such that 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.
That is, if and are positive integers, then a set system of cardinality greater than 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.
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 . It is known that .  A 2021 paper by Alweiss, Lovett, Wu, and Zhang gives the best progress towards the conjecture, proving that for . A month after the release of the first version of their paper, Rao sharpened the bound to .
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.
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 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.
- 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
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