the famous no-three-in-line problem by dudeney more than a century ago asks whether one can select 2n points from the grid $[n]^2$ such that no three are collinear. we present two results related to this problem. first, we give a non-trivial upper bound for the maximum size of a set in $[n]^4$ such that no four are coplanar. second, we characterize the behavior of the maximum size of a subset such that no three are collinear in a random set of $\mathbb{f}_q^2$, that is, the plane over the finite field of order q. we discuss their proofs and related open problems.