Suppose that all lines defined by pairs of distinct elements in a subset of $\mathbb F_p^2$ have different slopes. How large can such a subset be asymptotically (for primes $p\rightarrow \infty$)?
Choosing such a subset $\mathcal S$ at random, the anniversary theorem suggests that one can expect a repeated slope for $\mathcal S$ roughly of size $p^{1/4}$. Can one improve the exponent substantially for the largest possible subset $\mathcal S$?
Surely, we have $|S|\leq O(\sqrt p)$, as the number of slopes cannot exceed $p+1$. This estimate is sharp, up to a multiplicative constant.
Indeed, let $A$ be a Sidon set in $\mathbb F_p$, that is --- a set satisfying $a+b=c+d\Rightarrow \{a,b\}=\{c,d\}$ whenever $a,b,c,d\in S$. There exists such a set with $|A|=\Omega(\sqrt p)$.
Now let $S=\{(a,a^2)\colon a\in A\}\subset \mathbb F_p^2$. For any two distinct points $(a,a^2),(b,b^2)\in S$ the slope of the line determined by them is $(b^2-a^2)/(b-a)=a+b$, so all the slopes are distinct.
