The number of independent sets in regular bipartite expander graphs can be efficiently approximated by expressing it as the partition function of a suitable polymer model and truncating its cluster expansion. While this approach has been extensively used for graphs, surprisingly little is known about analogous questions in the context of hypergraphs. In this work, we apply this method to asymptotically determine the number of independent sets in regular $k$-partite $k$-uniform hypergraphs which satisfy natural expansion properties. The resulting formula depends only on the local structure of the hypergraph, making it computationally efficient. In particular, we provide a simple closed-form expression for linear hypergraphs.

翻译：正则二分扩张图中独立集的数量可通过将其表述为适当聚合物模型的配分函数并截断其簇展开来高效近似。尽管该方法已在图论中得到广泛应用，但令人惊讶的是，在超图背景下类似问题的研究却极为有限。本工作中，我们将此方法应用于满足自然扩张性质的正则$k$-部$k$-一致超图中独立集数量的渐近确定。所得公式仅依赖于超图的局部结构，因而具有计算高效性。特别地，我们为线性超图给出了简洁的闭式表达式。