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What is the max number of points in R^3, interconnected by generic curves?

Given a set of points connected by edges lying on an euclidean plane, I'd like to find which is the smaller dimension of the euclidean space where the graph can lie without an overlapping of the edges. Is it a standard problem? Which mathematical tools I have to know to manage with this kind of problems? I can obviously guess that I could always take $d=v-1$ where $v$ is the number of verticies but I can't understand which is the smaller dimension.

Thanks