In the paper Hardness of embedding simplicial complexes in $\mathbb{R}^{d}$ the abstract states that a finite simplicial complex of dimension $k$ embeds in $\mathbb{R}^{2k}$ while on page $856$ he says that it does not embed in $\mathbb{R}^{2k}$ but rather in $\mathbb{R}^{2k+1}$.
Which one is correct?
My understanding from the definition of a simplex $\Delta^k = \left\{(t_0,\cdots,t_k)\in\mathbb{R}^{k+1}\mid\Sigma_{i = 0}^{k}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}$
is that a finite simplicial complex of dimension $k$ embeds in $\mathbb{R}^{k+1}$, am I right?