Existential definability over the subword ordering

. We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the sigma 1 (i.e., existential) fragment is undecidable, already for binary alphabets A.However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable.We show that if |A| >= 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments sigma i: If |A| >= 3, then a relation is definable in sigma i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the sigma i-fragments for i >= 2 of the pure logic, where the words of A* are not available as constants.