Throughout, by "logic" I mean regular logic (in the sense of Ebbinghaus–Flum–Thomas) whose sentences are coded by elements of $\mathsf{HC}$. Say that $\mathcal{L}$ is **Barwise compact** iff whenever $\mathbb{A}$ is a countable admissible set and $X\subseteq\mathbb{A}$ is a $\mathbb{A}$-c.e. $\mathcal{L}$-theory each of whose $\mathbb{A}$-finite subtheories is satisfiable, then $X$ is satisfiable. Finally, say that a logic $\mathcal{L}$ has the **single-sentence DLS property** iff every satisfiable $\mathcal{L}$-sentence has a countable model.


It's easy to show that $\mathcal{L}\_{\omega\_1,\omega}$ has the single-sentence DLS property *(favorite silly proof: apply Mostowski absoluteness to $\mathit{Col}(\omega,\kappa)$ for large enough $\kappa$)*, and Barwise showed that $\mathcal{L}\_{\omega\_1,\omega}$ is Barwise compact. Moreover, Lindstrom's characterization (see [Väänänen, *Lindstrom's Theorem*](http://www.math.helsinki.fi/logic/opetus/lt/lindstrom_theorem1.pdf)) of first-order logic can be [tweaked](https://mathoverflow.net/questions/350778/improving-a-lindstrom-y-fact-about-mathcall-omega-1-omega) to show that no logic with the above two properties can be *much stronger than* $\mathcal{L}\_{\omega\_1,\omega}$. However, there's still a large gap here, and I'm curious whether anything is known about it. In particular:



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> Is there an already-introduced logic which is properly stronger than $\mathcal{L}\_{\omega\_1,\omega}$, has the single-sentence DLS property, and is Barwise compact?
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I'd also be interested in what happens if we weaken the definition of Barwise compactness to allow replacing "admissible set" with "transitive model of $T$" for some first-order $T\subseteq \mathit{Th}(\mathsf{HC})$.



Natural strong logic with Barwise compactness property
A very nice family of examples is provided by Harrington's 1980 paper *[Extensions of countable infinitary logic which preserve most of its nice properties](https://link.springer.com/content/pdf/10.1007/BF02021129.pdf)*. Harrington shows that if we expand $\mathcal{L}\_{\omega\_1,\omega}$ by any infinitary propositional connective satisfying an appropriate tameness property - and there are $2^{2^{\aleph\_0}}$ of these - we preserve multiple tameness properties, including Barwise compactness and dLS for individual sentences (the latter basically trivially).


