Definable maximal cofinitary groups of intermediate size

Using almost disjoint coding, we show that for each $1<M<N<\omega$ consistently $\mathfrak{d}=\mathfrak{a}_g=\aleph_M<\mathfrak{c}=\aleph_N$, where $\mathfrak{a}_g=\aleph_M$ is witnessed by a $\Pi^1_2$ maximal cofinitary group.Comment: 22 page

Defining Recursive Predicates in Graph Orders

We study the first order theory of structures over graphs i.e. structures of the form ($\mathcal{G},\tau$) where $\mathcal{G}$ is the set of all (isomorphism types of) finite undirected graphs and $\tau$ some vocabulary. We define the notion of a recursive predicate over graphs using Turing Machine recognizable string encodings of graphs. We also define the notion of an arithmetical relation over graphs using a total order $\leq_t$ on the set $\mathcal{G}$ such that ($\mathcal{G},\leq_t$) is isomorphic to ($\mathbb{N},\leq$). We introduce the notion of a \textit{capable} structure over graphs, which is one satisfying the conditions : (1) definability of arithmetic, (2) definability of cardinality of a graph, and (3) definability of two particular graph predicates related to vertex labellings of graphs. We then show any capable structure can define every arithmetical predicate over graphs. As a corollary, any capable structure also defines every recursive graph relation. We identify capable structures which are expansions of graph orders, which are structures of the form ($\mathcal{G},\leq$) where $\leq$ is a partial order. We show that the subgraph order i.e. ($\mathcal{G},\leq_s$), induced subgraph order with one constant $P_3$ i.e. ($\mathcal{G},\leq_i,P_3$) and an expansion of the minor order for counting edges i.e. ($\mathcal{G},\leq_m,sameSize(x,y)$) are capable structures. In the course of the proof, we show the definability of several natural graph theoretic predicates in the subgraph order which may be of independent interest. We discuss the implications of our results and connections to Descriptive Complexity

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 ?? (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 ?_i: If |A| ? 3, then a relation is definable in ?_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 ?_i-fragments for i ? 2 of the pure logic, where the words of A^* are not available as constants