Difference between revisions of "The Erdos-Rado sunflower lemma"

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<B>Disproven for <math>k=3,r=3</math></B>: set <math>|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.
 
<B>Disproven for <math>k=3,r=3</math></B>: set <math>|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.
  
 +
== 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>k</math> fixed we have <math>f(k,r)=k^r+o(k^r)</math> from Kostochka-Rödl-Talysheva.
 +
 +
{| class="wikitable"
 +
! style="font-weight: bold; font-style: italic;" | r\k
 +
! style="text-align: center; font-style: italic;" | 2
 +
! style="text-align: center; font-style: italic;" | 3
 +
! style="text-align: center; font-style: italic;" | 4
 +
! style="text-align: center; font-style: italic;" | 5
 +
! style="text-align: center; font-style: italic;" | 6
 +
! style="text-align: center; font-weight: bold; font-style: italic;" | k
 +
|-
 +
| style="text-align: center; font-style: italic;" | 3
 +
| style="text-align: right; font-weight: bold;" | 6
 +
| style="text-align: right; font-weight: bold;" | 20
 +
| style="text-align: right;" | 54-
 +
| style="text-align: right;" | 160-
 +
| style="text-align: right;" | 600-
 +
| style="text-align: center;" | ~3.16^k
 +
|-
 +
| style="text-align: center; font-style: italic;" | 4
 +
| style="text-align: right; font-weight: bold;" | 10
 +
| style="text-align: right;" | 38-
 +
| style="text-align: right;" | 114-
 +
| style="text-align: right;" | 380-
 +
| style="text-align: right;" | 1444-
 +
| style="text-align: center;" | ~3.36^k
 +
|-
 +
| style="text-align: center; font-style: italic;" | 5
 +
| style="text-align: right; font-weight: bold;" | 20
 +
| style="text-align: right;" | 88-
 +
| style="text-align: right;" | 400-
 +
| style="text-align: right;" | 1760-
 +
| style="text-align: right;" | 8000-
 +
| style="text-align: center;" | ~4.24^k
 +
|-
 +
| style="text-align: center; font-style: italic;" | 6
 +
| style="text-align: right; font-weight: bold;" | 27
 +
| style="text-align: right;" | 146-
 +
| style="text-align: right;" | 730-
 +
| style="text-align: right;" | 3942-
 +
| style="text-align: right;" | 21316-
 +
| style="text-align: center;" | ~5.26^k
 +
|}
 
== Threads ==
 
== Threads ==
  
Line 78: Line 125:
 
* An intersection theorem for systems of sets, A. V. Kostochka, Random Structures and Algorithms, 9 (1996), 213-221.
 
* An intersection theorem for systems of sets, A. V. Kostochka, Random Structures and Algorithms, 9 (1996), 213-221.
 
* [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on <math>\Delta</math>-systems], A. V. Kostochka
 
* [http://www.math.uiuc.edu/~kostochk/docs/2000/survey3.pdf Extremal problems on <math>\Delta</math>-systems], A. V. Kostochka
 +
* On Systems of Small Sets with No Large Δ-Subsystems, A. V. Kostochka, V. Rödl, and L. A. Talysheva, Comb. Probab. Comput. 8 (1999), 265-268.
 
* [http://arxiv.org/abs/1202.4196 On Erdos' extremal problem on matchings in hypergraphs], T. Luczak, K. Mieczkowska
 
* [http://arxiv.org/abs/1202.4196 On Erdos' extremal problem on matchings in hypergraphs], T. Luczak, K. Mieczkowska
 
* Intersection theorems for systems of sets,  J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.
 
* Intersection theorems for systems of sets,  J. H. Spencer, Canad. Math. Bull. 20 (1977), 249-254.

Revision as of 17:33, 28 November 2015

The problem

A sunflower (a.k.a. Delta-system) of size [math]r[/math] is a family of sets [math]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]f(k,r)[/math] so that every family [math]\cal F[/math] of [math]k[/math]-sets with more than [math]f(k,r)[/math] members contains a sunflower of size [math]r[/math].

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

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

[math]\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]\alpha \lt 1[/math], and some [math]D(r,\alpha)[/math] depending on [math]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 f(k,r) \leq C^k [/math]

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

Variants and notation

Given a family F of sets and a set S, the star of S is the subfamily of those sets in F 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]f(k,r;m,n) \leq C_r^k n^{k-m}[/math] for some [math]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]f(k,2;1,n) = \binom{n-1}{k-1}[/math] (1)

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

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

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

for [math]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 k!/k^k|F|.

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]k=3,r=3[/math]: set [math]|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.

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]k[/math] fixed we have [math]f(k,r)=k^r+o(k^r)[/math] from Kostochka-Rödl-Talysheva.

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!