  For a given number of colors, $s$, the guessing number of a graph is the
(base $s$) logarithm of the cardinality of the largest family of colorings of
the vertex set of the graph such that the color of each vertex can be
determined from the colors of the vertices in its neighborhood. This quantity
is related to problems in network coding, circuit complexity and graph entropy.
We study the guessing number of graphs as a graph property in the context of
classic extremal questions, and its relationship to the forbidden subgraph
property. We find the extremal number with respect to the property of having
guessing number $\leq a$, for fixed $a$. Furthermore, we find an upper bound on
the saturation number for this property, and a method to construct further
saturated graphs that lie between these two extremes. We show that, for a fixed
number of colors, bounding the guessing number is equivalent to forbidding a
finite set of subgraphs.
