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Abstract

A first order structure $\mathfrak M$ with universe M is atomic compact if every system of atomic formulas with parameters in $M$ is satisfiable in $\mathfrak M$ provided each of its finite subsystems is. We consider atomic compactness for the class of reflexive (symmetric) graphs. In particular, we investigate the extent to which "sparse" graphs (i.e. graphs with "few" vertices of "high" degree) are compact with respect to systems of atomic formulas with "few" unknowns, on the one hand, and are pure restrictions of their Stone-Čech compactifications, on the other hand.