More canonical forms and dense free subsets

AbstractAssuming the existence of ω compact cardinals in a model on GCH, we prove the consistency of some new canonization properties on ℵω. Our aim is to get as dense patterns in the distribution of indiscernibles as possible. We prove Theorem 2.1. thm2.1Suppose the consistency of “ZFC+GCH+ there are infinitely many compact cardinals”. Then the following is consistent: ZFC+GCH+ and for every family (fn)0<n<w of functions on ℵw such that fn is n-ary and regressive, there are sets Sn,0<n<w, such that for all 0<n<w,Sn⊆[ℵn,ℵn+1,|Sn|⩾ℵn−1, and fn↾(∏i=1nSi) is constant.We generalize this to higher arities, and find that the following is consistent relatively to the same large cardinal assumptions: Given a family of regressive functions (fn)0<n<ω on ℵω and a function r:ω→ω, there is a family of sets (Sn)0<n<ω that have a certain size and that are indiscernible for values under fn for all 0<n<ω simultaneously, if fn picks r(m) increasing arguments from Sm for 0