The minimum number of ee-vertex-covers among hypergraphs with ee edges of given ranks

We study the problem that among all hypergraphs with ee edges of ranks l1,…,lel1,…,le and vv vertices, which hypergraph has the least number of vertex-covers of size ee. The problem is very difficult and we only get some partial answers. We show an application of our results to improve the error-tolerance of a pooling design proposed in the literature.