Ramsey Methods and the problem DU

We consider Fremlin’s notion of 1/2-density and the related notion of Fremlin cardinals. A well known related question is if every 1/2-dense hereditary family on an uncountable cardinal must have an infinite homogeneous family. These notions do not seem to lend themselves to Ramseyan methods. In particular, it is not known if a Fremlin cardinal must be a large cardinal. We introduce a related notion of 1/2-dense cardinals which is easier to handle using Ramsey methods. We show that a 1/2-dense cardinal must be at least strongly inaccessible. On the other hand, David Aspero showed that an ω-Erd¨os cardinal must be 1/2-dense.