$\omega$-cohesive sets

Abstract:

We define and investigate $\omega$-cohesiveness, a strong notion of indecomposability for subsets of the integers and their isols. This notion says, for example, that if $X$ is the isol of an $\omega$-cohesive set then, for any integer $n$ implies that, for some integer $k, \cdot (\begin {array}{*{20}{c}} {X - k} \\ n \\ \end {array} ) \leq Y$ or $Z$. From this it follows that if $f(x) \in {T_1}$, the collection of almost recursive combinatorial polynomials, then the predecessors of ${f_\Lambda }(X)$ are limited to isols ${g_\Lambda }(X)$ where $g(X) \in {T_1}$. We show existence of $\omega$-cohesive sets. And we show that the isol of an $\omega$-cohesive set is an $n$-order indecomposable isol as defined by Manaster. This gives an alternate proof to one half of Ellentuck’s theorem showing a simple algebraic difference between the isols and cosimple isols. In the last section we study functions of several variables when applied to isols of $\omega$-cohesive sets.