Count( qq) versus the pigeon-hole principle

For each p ≤ 2 there exist a model M∗ of I∆0(α) which satisfies the Count(p) principle. Furthermore if p contain all prime factors of q there exist n, r ∈ M∗ and a bijective map f ∈ Set(M∗) mapping {1, 2, ..., n} onto {1, 2, ..., n+ qr}. A corollary is a complete classification of the Count(q) versus Count(p) problem. Another corollary solves an open question ([3]). In this note I state and prove a Theorem which actually can be viewed as the main result of [9]. Theorem: Suppose that r(n) is an function with (a) limn→∞ r(n) =∞. (b) For all > 0 limn→∞ q r(n) n = 0 For each q, p ≥ 2 Count(p) 6` PHP∗+qr(∗)(bij) if p divides a power of q. Here PHP∗+s(bij) is the the elementary principle stating that there does not exists n and a bijective map from {1, 2, ..., n} onto {1, 2, ..., n+ s}.And Count(p) is the elementary matching principle stating that if {1, 2, ..., n} is divided into disjoint p-element subsets, then p divides n. Proof: As in [9] let M be a countable non-standard model of first order Arithmetic. Then by a similar forcing construction (which actually avoids ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.