More canonical forms and dense free subsets

Assuming the existence of ! compact cardinals in a model on GCH, we prove the consistency of some new canonization properties on א!. Our aim is to get as dense patterns in the distribution of indiscernibles as possible. We prove Theorem 2.1. Theorem 2.1. Suppose the consistency of “ZFC + GCH + there are in(nitely 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 (∏ni=1 Si) is constant. We generalize this to higher arities, and >nd that the following is consistent relatively to the same large cardinal assumptions: Given a family of regressive functions (fn)0¡n¡! on א! and a function r :! → !, there is a family of sets (Sn)0¡n¡! that have a certain size and that are indiscernible for values under fn for all 0¡n¡! simultaneously, if fn picks r(m) increasing arguments from Sm for 0¡m6 n. We determine the locations of the sets Sn in א!. This, together with some additional work on indiscernibility over as many smaller parameters as possible, yields the consistency of the existence of free subsets with at least one point in every in>nite cardinal interval of א!. c © 2003 Elsevier B.V. All rights reserved. MSC: 03E02; 03E35; 03E55