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 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 supermultiplicative, i.e., $f(ab)\ge f(a)f(b)$.

UPDATE. Best lower bound for $C_3$ that I am aware of is $\sqrt 5$ which follows from $f(2)=5$, this is exactly the same construction which is described in Douglas's comment.