How many facets can a 0-1 polytope with $n$ vertices in $R^d$ have?

Let $f(d)$ denote the maximum number of facets of a 0-1 polytope in $R^d$. The question such as ``is this function bounded by an exponential in $d$?'' was open just until recently. The negative answer was given by Bárány and Pór who proved the superexponential behavior of $f(d)$.

Theorem 6 (Bárány and Pór [BP00])   There is a positive constant $c$ such that

$$f(d) > \left ( \frac{c\;d}{\log d} \right )^\frac{d}{4}.$$ (2)

This is a recent breakthrough in the theory of 0-1 polytopes.