# The state of art of the sunflower lemma

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

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With a finite universe you can take $c_{|U|} > 2^{|U|}$, making the statement vacuous since no $F$ satisfy the condition. –  Ben Barber Dec 1 '12 at 17:10
Nice question. The best result I know is due to Kostochka: For all $m>1$ there is a $c_m$ such that for any $n$, an $n$-uniform family $F$ not admitting a $3$-sunflower system and of largest possible cardinality satisfies $|F|\le c_m n! m^{-n}$. –  Andres Caicedo Dec 1 '12 at 17:15
@Andres could you give a link? I have got some Kostochka's papers. ButI might not have this one. –  WangYao Dec 1 '12 at 18:06
@Ben. Yes.But since so $c_{|U|}>2^{|U|}$ is an trivial bound.We need something better. –  WangYao Dec 1 '12 at 18:14
Wang, This is a few years old, so better results may be known. The reference I have is: MR1425216 (98d:05145) Alexandr V. Kostochka. A bound of the cardinality of families not containing $\Delta$-systems, in The mathematics of Paul Erdős, II, 229–235, Algorithms Combin., 14, Springer, Berlin, 1997. –  Andres Caicedo Dec 1 '12 at 20:24