K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces $X$ defined over a number field $k$, if there is an algebraic point on $X$ that lies on only finitely many rational curves, then there is an algebraic point on $X$ that lies on no rational curves. In particular, our theorem applies to a large class of elliptic $K3$ surfaces, which relates to a question posed by Bogomolov in 1981. We apply our results to construct an explicit algebraic point on a $K3$ surface that does not lie on any smooth rational curves.