Given a finite set $A$ with $n$ elements, what would be a good upper bound for the size of a largest collection $\mathcal{F}$ of subsets of $A$ which satisfy the following condition: Any two elements of $\mathcal{F}$ have at most one element of $A$ in common.
Following advice I got on #math, I tried feeding oeis with the values of this count for the first few values of $n$ (namely: $1, 2, 4, 7, 11$), but in the results there I couldn't find any reference to the kind of thing I am interested in.
Would you know of good upper bounds for this? Where/what should I look up?