A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = Z $ for all $i \neq j$ for some fixed $Z$. A well-known conjecture of Erdos and Rado says that there is a constant $C_t$ such that in any $k$-uniform family of size at least $C_t^k$ there is a $t$-sunflower. This is still wide open even for $t=3$, for more see http://en.wikipedia.org/wiki/Sunflower_(mathematics).

My question is, what is the best lower bound for $C_3$? So what is the largest known example of a $k$-uniform family that does not have a $3$-sunflower?

We can also study this as some function $f$ of $k$. I am even interested in small values, like up to $20$, if anyone can compute it. It is easy to see that $f$ is logsuperadditive. In case this is not a word, I mean $f(a+b)\ge f(a)f(b)$.

UPDATEs. Best lower bound for $C_3$ is $\sqrt[3]{20}\approx 2.714$, which follows from $f(3)\ge20$, see the comment posted by Douglas.

We also know $f(1)=2, f(2)=6, 20\le f(3)\le 32$, these were obtained by Mirko (see his comment) using $f(k)\le 2kf(k-1)-2k+2$.