  An efficient implicit representation of an $n$-vertex graph $G$ in a family
$\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length
$O(\log n)$ so that the adjacency between every pair of vertices can be
determined only as a function of their codes. This function can depend on the
family but not on the individual graph. Every family of graphs admitting such a
representation contains at most $2^{O(n\log(n))}$ graphs on $n$ vertices, and
thus has at most factorial speed of growth.
  The Implicit Graph Conjecture states that, conversely, every hereditary graph
family with at most factorial speed of growth admits an efficient implicit
representation. We refute this conjecture by establishing the existence of
hereditary graph families with factorial speed of growth that require codes of
length $n^{\Omega(1)}$.
