In the Euclidean plane (though I assume the following result can be generalized to any Euclidean n-space), Tarski showed that one can define what it is to be a straight line solely in terms of the betweenness relations among the points on that line. As @achille hui notes, according to Tarski, 'betweenness' is a triadic relation. The atomic sentence Bxyz denotes that x is between y and z, which intuitively means that x is a point of the line segment yz.
In a spacetime of constant positive curvature (i.e., a spherical 3-space, plus a fourth temporal dimension), straight lines are replaced by geodesics. Analogously to what Tarski showed in the Euclidean case, I'm wondering whether one can define what it is to be a geodesic solely in terms of some sort of betweenness relations among the points on that geodesic? (If this is possible, I'm assuming the betweenness relation in the spacetime of constant positive curvature will be distinct from Tarski's Euclidean betweenness relation.)