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A classical result in topology for which I can't find a reference for is that a simplicial complex $K$ of dimension $d$ with $n$ vertices can be linearly embedded into $\mathbb{R}^{k}$ when $k=2d+1$. Does anyone know if an example of such a linear embedding can be given explicitly (or determined efficiently)?

If not, what is the smallest $k\leq n$ for which a general explicit construction is known (keeping $n$ and $d$ fixed)?

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Consider the curve $C$ given by $(t,t^2, \ldots, t^{2d+1})$ in ${\mathbb R}^{2d+1}$. Any collection of distinct $2d+2$ points on this curve is in general position. In particular, the two $d$-dimensional affine subspaces of ${\mathbb R}^{2d+1}$ determined by any two distinct collections of $d+1$ points on this curve are disjoint. Given a simplicial $d$-complex $K$, map its vertices to distinct points on the curve $C$, and extend linearly to each simplex. By the general position property of the curve $C$, this is an embedding of $K$ into ${\mathbb R}^{2d+1}$.

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  • $\begingroup$ Hi Slava. Thank you. Do you know of a citation? $\endgroup$ – Anon Y. Mouse Jul 4 '16 at 21:08
  • $\begingroup$ The general position property of points on this curve is often used in Ramsey theory; in algebraic geometry the equation defines the rational normal curve. (I haven't come across a reference that explicitly states the answer to your question about a specific embedding of simplicial complexes.) $\endgroup$ – Slava Krushkal Jul 4 '16 at 21:27
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    $\begingroup$ @AnonY.Mouse See cyclic polytopes. en.wikipedia.org/wiki/Cyclic_polytope $\endgroup$ – Douglas Zare Jul 4 '16 at 22:10

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