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)

A bound of the cardinality of families not containing $\Delta$-systems, inThe mathematics of Paul Erdős, II, 229–235, Algorithms Combin., 14, Springer, Berlin, 1997. – Andrés E. Caicedo Dec 1 '12 at 20:24