A variant of the Hales–Jewett theorem
Abstract
It was shown by Bergelson that any set B ⊆ ℕ with positive upper multiplicative density contains nicely intertwined arithmetic and geometric progressions: for each k ∈ ℕ there exist a , b , d ∈ ℕ such that { b ( a + id ) j : i , j ∈ {1, 2, …, k }}⊆ B . In particular, one cell of each finite partition of ℕ contains such configurations. We prove a Hales–Jewett-type extension of this partition theorem.