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.

翻译：本文研究类型图，即由联合类型导出的二分图。我们考察该图中具有指数规模（以类型长度$n$为指数）顶点数目的诱导二分子图的最大边密度。这可以看作一个等周问题。当类型长度趋于无穷时，我们给出了最大边密度指数的渐近精确界。同时研究了类型图的二部团率区域，定义为存在二部团且两侧顶点数分别为$2^{nR_{1}}$和$2^{nR_{2}}$的$(R_{1},R_{2})$集合，并给出了该区域的渐近精确界。随后讨论这些结果与非交互模拟及超压缩性不等式的关联。此外，作为应用，我们为二进制加法信道零误差容量区域给出了新的外边界，改进了Austrin、Kaski、Koivisto和Nederlof此前的最优已知界。本文证明基于类型方法与线性代数。