The Erdos-Rado sunflower lemma: Difference between revisions

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Below is a collection of known constructions for small values, taken from Abbott-Exoo.
Below is a collection of known constructions for small values, taken from Abbott-Exoo.
Boldface stands for matching upper bound (and best known upper bounds are planned to be added to other entries).
Boldface stands for matching upper bound (and best known upper bounds are planned to be added to other entries).
Also note that for <math>k</math> fixed we have <math>f(k,r)=k^r+o(k^r)</math> from Kostochka-Rödl-Talysheva.
Also note that for <math>k</math> fixed we have <math>f(k,r)=r^k+o(r^k)</math> from Kostochka-Rödl-Talysheva (note that that paper swaps the notation for k and r).


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Latest revision as of 13:26, 29 July 2020

The problem

A sunflower (a.k.a. Delta-system) of size [math]\displaystyle{ r }[/math] is a family of sets [math]\displaystyle{ A_1, A_2, \dots, A_r }[/math] such that every element that belongs to more than one of the sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

Erdos-Rado Delta-system theorem: There is a function [math]\displaystyle{ f(k,r) }[/math] so that every family [math]\displaystyle{ \cal F }[/math] of [math]\displaystyle{ k }[/math]-sets with more than [math]\displaystyle{ f(k,r) }[/math] members contains a sunflower of size [math]\displaystyle{ r }[/math].

(We denote by [math]\displaystyle{ f(k,r) }[/math] the smallest integer that suffices for the assertion of the theorem to be true.) The simple proof giving [math]\displaystyle{ f(k,r)\le k! (r-1)^k }[/math] can be found here.

The best known general upper bound on [math]\displaystyle{ f(k,r) }[/math] (in the regime where [math]\displaystyle{ r }[/math] is bounded and [math]\displaystyle{ k }[/math] is large) is

[math]\displaystyle{ \displaystyle f(k,r) \leq D(r,\alpha) k! \left( \frac{(\log\log\log k)^2}{\alpha \log\log k} \right)^k }[/math]

for any [math]\displaystyle{ \alpha \lt 1 }[/math], and some [math]\displaystyle{ D(r,\alpha) }[/math] depending on [math]\displaystyle{ r,\alpha }[/math], proven by Kostkocha from 1996. The objective of this project is to improve this bound, ideally to obtain the Erdos-Rado conjecture

[math]\displaystyle{ \displaystyle f(k,r) \leq C^k }[/math]

for some [math]\displaystyle{ C=C(r) }[/math] depending on [math]\displaystyle{ r }[/math] only. This is known for [math]\displaystyle{ r=1,2 }[/math](indeed we have [math]\displaystyle{ f(k,r)=1 }[/math] in those cases) but remains open for larger r.

Variants and notation

Given a family [math]\displaystyle{ \cal F }[/math] of sets and a set S, the star of S is the subfamily of those sets in [math]\displaystyle{ \cal F }[/math] containing S, and the link of S is obtained from the star of S by deleting the elements of S from every set in the star. (We use the terms link and star because we do want to consider eventually hypergraphs as geometric/topological objects.)

We can restate the delta system problem as follows: f(k,r) is the maximum size of a family of k-sets such that the link of every set A does not contain r pairwise disjoint sets.

Let f(k,r;m,n) denote the largest cardinality of a family of k-sets from {1,2,…,n} such that that the link of every set A of size at most m-1 does not contain r pairwise disjoint sets. Thus f(k,r) = f(k,r;k,n) for n large enough.

Conjecture 1: [math]\displaystyle{ f(k,r;m,n) \leq C_r^k n^{k-m} }[/math] for some [math]\displaystyle{ C_r }[/math] depending only on r.

This conjecture implies the Erdos-Ko-Rado conjecture (set m=k). The Erdos-Ko-Rado theorem asserts that

[math]\displaystyle{ f(k,2;1,n) = \binom{n-1}{k-1} }[/math] (1)

when [math]\displaystyle{ n \geq 2k }[/math], which is consistent with Conjecture 1. More generally, Erdos, Ko, and Rado showed

[math]\displaystyle{ f(k,2;m,n) = \binom{n-m}{k-m} }[/math]

when [math]\displaystyle{ n }[/math] is sufficiently large depending on k,m. The case of smaller n was treated by several authors culminating in the work of Ahlswede and Khachatrian.

Erdos conjectured that

[math]\displaystyle{ f(k,r;1,n) = \max( \binom{rk-1}{k}, \binom{n}{k} - \binom{n-r}{k} ) }[/math]

for [math]\displaystyle{ n \geq rk }[/math], generalising (1), and again consistent with Conjecture 1. This was established for k=2 by Erdos and Gallai, and for r=3 by Frankl (building on work by Luczak-Mieczkowska).

A family of k-sets is balanced (or k-colored) if it is possible to color the elements with k colors so that every set in the family is colorful.

Reduction (folklore): It is enough to prove Erdos-Rado Delta-system conjecture for the balanced case.

Proof: Divide the elements into d color classes at random and take only colorful sets. The expected size of the surviving colorful sets is [math]\displaystyle{ k!/k^k \cdot |\cal F| }[/math].

Hyperoptimistic conjecture: The maximum size of a balanced collection of k-sets without a sunflower of size r is (r-1)^k.

Disproven for [math]\displaystyle{ k=3,r=3 }[/math]: set [math]\displaystyle{ |V_1|=|V_2|=|V_3|=3 }[/math] and use ijk to denote the 3-set consisting of the i^th element of V_1, j^th element of V_2, and k^th element of V_3. Then 000, 001, 010, 011, 100, 101, 112, 122, 212 is a balanced family of 9 3-sets without a 3-sunflower.

A weak sunflower (weak Delta-system) of size [math]\displaystyle{ r }[/math] is a family of [math]\displaystyle{ r }[/math] sets, [math]\displaystyle{ A_1,\ldots,A_r }[/math], such that their pairwise intersections have the same size, i.e., [math]\displaystyle{ |A_i\cap A_j|=|A_{i'}\cap A_{j'}| }[/math] for every [math]\displaystyle{ i\ne j }[/math] and [math]\displaystyle{ i'\ne j' }[/math]. If we denote the size of the largest family of [math]\displaystyle{ k }[/math]-sets without an [math]\displaystyle{ r }[/math]-weak sunflower by [math]\displaystyle{ g(k,r) }[/math], by definition we have [math]\displaystyle{ g(k,r)\le f(k,r) }[/math]. For this function we also know that [math]\displaystyle{ g(a+b,r)\ge g(a,r)g(b,r) }[/math], as shown by Abbott-Hanson in On finite Δ-systems II.

Also, if we denote by [math]\displaystyle{ R_r(k)-1 }[/math] the size of the largest complete graph whose edges can be colored with [math]\displaystyle{ r }[/math] colors such that there is no monochromatic clique on [math]\displaystyle{ k }[/math] vertices, then we have [math]\displaystyle{ g(k,r)\le R_r(k)-1 }[/math], as we can color the edges running between the [math]\displaystyle{ k }[/math]-sets of our weak sunflower-free family with the intersection sizes. For all three functions only exponential lower bounds and factorial type upper bounds are known.

Denote by [math]\displaystyle{ 3DES(n) }[/math] the largest integer such that there is a group of size [math]\displaystyle{ n }[/math] and a subset [math]\displaystyle{ S }[/math] of size [math]\displaystyle{ 3DES(n) }[/math] without three disjoint equivoluminous subsets, i.e., there is no [math]\displaystyle{ S=S_1\cup^* S_2\cup^* S_3\cup^* S_{rest} }[/math] such that [math]\displaystyle{ \sum_{s\in S_1} s=\sum_{s\in S_2} s=\sum_{s\in S_3} s }[/math]. As noticed by Alon-Shpilka-Umans, [math]\displaystyle{ {3DES(n) \choose DES(n)} / n \le g(DES(n),3) }[/math] holds, thus if [math]\displaystyle{ g(k,3) }[/math] grows exponentially, then [math]\displaystyle{ 3DES(n)=O(\log n) }[/math].

Small values

Below is a collection of known constructions for small values, taken from Abbott-Exoo. Boldface stands for matching upper bound (and best known upper bounds are planned to be added to other entries). Also note that for [math]\displaystyle{ k }[/math] fixed we have [math]\displaystyle{ f(k,r)=r^k+o(r^k) }[/math] from Kostochka-Rödl-Talysheva (note that that paper swaps the notation for k and r).

r\k 2 3 4 5 6 ...k
3 6 20 54- 160- 600- ~3.16^k
4 10 38- 114- 380- 1444- ~3.36^k
5 20 88- 400- 1760- 8000- ~4.24^k
6 27 146- 730- 3942- 21316- ~5.26^k

Threads

External links

Bibliography

Edits to improve the bibliography (by adding more links, Mathscinet numbers, bibliographic info, etc.) are welcome!

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