The state of art for sunflower system

I am interesting in the sunflower system and its applications in computer science.

Given a Universe $U$ and a collection of $k$ sets $A_i$ is called a k-sunflower system if $A_i \cap A_j = Y$ for all $i \neq j$. And $Y$ is called as the core and $A_i - Y$ is called petals.

A family of sets $F$ is called $s$-uniform is all the sets it contained possess $s$ elements.

Erdos and Rado proved that for a $s$ uniform family of sets $F$ , $F$ must contain a $k$-sunflower system petals if $|F| > s!(k-1)^s$.

This result is called the sunflower lemma and has many important applications.

Erdos conjectured that for every $k$ there exist a constant $c_k$ such that the upper bound should be $c_k^s$ every $s$-uniform family $F$. (The sunflower conjecture)

Unfortunately, this conjecture is still open for $k=3$.

Here is what I want to know.

If we limit the number of elements in the universe $U$.Suppose $|U|$= $u$. Then the problem turns out be:

Given a universe with $u$ elements, and $s$-uniform family $F$ of sets containing the elements in $U$, we supposed can find sequence of constants $c_1$, $c_2$, $c_3$ ,... such that every $s$-uniform family $F$ contains a $3$-sunflower system if $|F|>$ $c_i^s$ and $|U|=i$.

Moreover, if we could prove that the sequence $c_i$ converges to a constant $c$, then it seems we can prove the sunflower conjecture.

But I cannot find such result.It might be that this approach is too stupid or too hard.

Could any one provide the state of art of sunflower lemma and the conjecture(finite version is also OK).

Here is some I can provide. There is a chapter in Junka's book The Extremal Combinatorics.

The paper above is one of its application(finite version)

On Sunflowers and Matrix Multiplication N Alon et.al

• there does not seem to be much direct work on it other than new applications & alons recent paper you cite, which might increase interest & is prob best place to start for refs (& juknas book is also unbeatable). here is a nice summary of the interconnections by kalai on his blog
– vzn
Dec 1 '12 at 21:42
• i think having a $c_i$ that depends on $i = |U|$ makes the problem trivial, as you can set $c_i = 2^i$. also my impression is that having no dependence on $|U|$ is the interesting thing about the lemma Dec 2 '12 at 20:23
• @SashoNikolov. Thanks for reply Yes, what we want is having no dependence on $|U|$. but if we have $|U|$, then we can explicitly build the maximal family $F$. What I wonder is that if this explicit building could show something interesting for the problem. For example can we find a family with $2^{i-\epsilon}$ that still does not contain a sunflower system. I tried to build such maximal family, but it seems so hard. I cannot make some larger maximal family other that the example on Junka's Book(Cha7). Dec 3 '12 at 17:10
• briefly, I am asking if we can improve the lower bound. Dec 3 '12 at 17:10