  In this paper, we study the type graph, namely, a bipartite graph induced by
a joint type. We investigate the maximum edge density of induced bipartite
subgraphs of this graph having a number of vertices on each side on an
exponential scale in the length $n$ of the type. This can be seen as an
isoperimetric problem. We provide asymptotically sharp bounds for the exponent
of the maximum edge density as the length of the type goes to infinity. We also
study the biclique rate region of the type graph, which is defined as the set
of $(R_{1},R_{2})$ such that there exists a biclique of the type graph which
has respectively $2^{nR_{1}}$ and $2^{nR_{2}}$ vertices on the two sides. We
provide asymptotically sharp bounds for the biclique rate region as well. We
then discuss the connections of these results to noninteractive simulation and
hypercontractivity inequalities. Furthermore, as an application of our results,
a new outer bound for the zero-error capacity region of the binary adder
channel is provided, which improves the previously best known bound, due to
Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on
the method of types and linear algebra.
