More on pooling spaces

A pooling space   is a ranked poset PP such that the subposet w+w+ induced by the elements above ww is atomic for each element ww of PP. Pooling spaces were introduced in [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Math. 282 (2004) 163–169] for the purpose of giving a uniform way to construct pooling designs, which have applications to the screening of DNA sequences. Many examples of pooling spaces were given in that paper. In this paper, we clarify a few things about the definition of pooling spaces. Then we find that a geometric lattice, a well-studied structure in literature, is also a pooling space. This provides us many classes of pooling designs, some old and some new. We study the pooling designs constructed from affine geometries. We find that some of them meet the optimal bounds related to a conjecture of Erdös, Frankl and Füredi.