Families close to disjoint ones (Q794639)

A system \(\{A_{\alpha}:\alpha<\lambda \}\) of sets, each of cardinality \(\mu\), is called sparse if there are sets \(B_{\alpha}\) with size less than \(\mu\) such that the system \(\{A_{\alpha}- B_{\alpha}:\alpha<\lambda \}\) is disjoint. It is shown that some results of E. W. Miller and Erdős-Hajnal on property B can be modified to get sparseness, which is obviously much stronger. It is also shown that if \(\kappa>\mu\) are regular and \(\kappa\) is weakly compact (or \(\mu =\omega\) and \(\kappa\) is supercompact) then there exists a generic extension in which \(\kappa\) becomes \(\mu^{++}\) and every non-sparse system of size \(\kappa\) (of any size) contains a non-sparse subsystem of size \(\mu^+.\) In recent unpublished work the author succeeded in extending the result between parentheses to the unrestricted case, and also in extending the Miller-type results up to a natural limit.