# 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 – Sasho Nikolov 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). – Yao Wang Dec 3 '12 at 17:10
• briefly, I am asking if we can improve the lower bound. – Yao Wang Dec 3 '12 at 17:10