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Martin Sleziak
Martin Sleziak
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A slight improvement on the lower bound was recently obtained by Alexey Glazyrin (http://arxiv.org/abs/0910.4200Lower bounds for the simplexity of the $n$-cube: https://arxiv.org/abs/0910.4200, http://www.sciencedirect.com/science/article/pii/S0012365X12003974https://doi.org/10.1016/j.disc.2012.09.002). Basically, it changes the $2^n$ factor in Nick's initial post to an $e^n$. Smith's hyperbolic volume method gave an $\sqrt{6}^n$.

But, of course, as Greg points out, the real open question is whether the simplicity of the cube grows as the $c^n n!/n^{n/2}$ of the lower bounds or as the $c^n n!$ that can be achieved by actual constructions (or something in between).

A slight improvement on the lower bound was recently obtained by Alexey Glazyrin (http://arxiv.org/abs/0910.4200, http://www.sciencedirect.com/science/article/pii/S0012365X12003974). Basically, it changes the $2^n$ factor in Nick's initial post to an $e^n$. Smith's hyperbolic volume method gave an $\sqrt{6}^n$.

But, of course, as Greg points out, the real open question is whether the simplicity of the cube grows as the $c^n n!/n^{n/2}$ of the lower bounds or as the $c^n n!$ that can be achieved by actual constructions (or something in between).

A slight improvement on the lower bound was recently obtained by Alexey Glazyrin (Lower bounds for the simplexity of the $n$-cube: https://arxiv.org/abs/0910.4200, https://doi.org/10.1016/j.disc.2012.09.002). Basically, it changes the $2^n$ factor in Nick's initial post to an $e^n$. Smith's hyperbolic volume method gave an $\sqrt{6}^n$.

But, of course, as Greg points out, the real open question is whether the simplicity of the cube grows as the $c^n n!/n^{n/2}$ of the lower bounds or as the $c^n n!$ that can be achieved by actual constructions (or something in between).

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Francisco Santos
Francisco Santos
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A slight improvement on the lower bound was recently obtained by Alexey Glazyrin (http://arxiv.org/abs/0910.4200, http://www.sciencedirect.com/science/article/pii/S0012365X12003974). Basically, it changes the $2^n$ factor in Nick's initial post to an $e^n$. Smith's hyperbolic volume method gave an $\sqrt{6}^n$.

But, of course, as Greg points out, the real open question is whether the simplicity of the cube grows as the $c^n n!/n^{n/2}$ of the lower bounds or as the $c^n n!$ that can be achieved by actual constructions (or something in between).