Exponential Triples

Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind a, b, ab (with a, b > 1). Also, we will show that any multiplicative IP ∗ set is an “exponential IP set”, the analogue of an IP set with respect to exponentiation. Introduction A well-known theorem by Hindman states that given any finite partition of N, there exists an infinite sets X and one cell of the partition containing the finite sums of X (and also the finite products of some infinite set Y ), see [Hi]. Ultrafilters can be used to give a simpler proof than the original one, see [Be], [HS]. We will be interested in similar results involving exponentiation instead of addition and multiplication, and our methods of proof will involve ultrafilter arguments. The first main result of this paper is the following. Theorem 1. Consider a partition of the natural numbers N = A ∪ B. Either A or B contains infinitely many triples a, b, a, with a, b > 1. Next, we will provide results (Theorems 14 and 15) which allow to find larger structures than the triples as above inside multiplicative IP ∗ sets (see Definition 12). A corollary of those theorems (Corollary 16) is given below. Definition 2. Consider an infinite set X ⊆ N and write X = {xi}i∈N, with xj < xj+1 for each j. Define inductively FE n+1(X) = {y n+1|y ∈ FE n(X)} ∪ FE I n(X) ∪ {xn+1}, FE n+1(X) = {(xn+1) |y ∈ FE n (X)} ∪ FE II n (X) ∪ {xn+1}, This is available on Bergelson’s webpage http://www.math.osu.edu/vitaly/ the electronic journal of combinatorics v18 (2011), #P147 1 with FE 0(X) = FE II 0 (X) = {x0}. Set FE(X) = ⋃ n∈N FE n(X), FE(X) = ⋃