Splitting families of sets, part 1

I will present several known results in the topic of splitting families of sets. Miller’s splitting theorem from 1937 was proven for finite n > 0 and ρ-uniform families where ρ is infinite. Under the assumptions of GCH and without the use of more modern tools, this approach is generalized here in order to prove several stronger theorems by Erdős and Hajnal.
Then a more modern approach by Kojman will be presented, where he used Shelah’s RGCH in order to eliminate the GCH from some of the theorems when ρ ≥ בω (ν). This was done using filtrations with respect to anti-monotone functions.