  The implicit representation conjecture concerns hereditary families of
graphs. Given a graph in such a family, we want to assign some string of bits
to each vertex in such a way that we can recover the information about whether
2 vertices are connected or not using only the 2 strings of bits associated
with those two vertices. We then want to minimise the length of this string.
The conjecture states that if the family is hereditary and small enough (it
only has $2^{O(n\ln(n))}$ graphs of size $n$), then $O(\ln(n))$ bits per vertex
should be sufficient. The trivial bounds on this problem are that: (1) some
families require at least $\ln_2(n)$ bits per vertex ; (2) $(n-1)/2+\ln_2(n)$
bits per vertex are sufficient for all families.
  In this paper, we will be talking about a special case of the implicit
representation conjecture, where the family is semi-algebraic (which roughly
means that the vertices are points in some euclidean space, and the edges are
defined geometrically, or according to some polynomials). We will first prove
that the `obvious' way of storing the information, where we store an
approximation of the coordinates of each vertex, doesn't work. Then we will
come up with a way of storing the information that requires $O(n^{1-\epsilon})$
bits per vertex, where $\epsilon$ is some small constant depending only on the
family. This is a slight improvement over the trivial bound, but is still a
long way from proving the conjecture.
