This paper investigates the class of k-universal finite graphs, a local
analog of the class of universal graphs, which arises naturally in the study of
finite variable logics. The main results of the paper, which are due to Shelah,
establish that the class of k-universal graphs is not definable by an infinite
disjunction of first-order existential sentences with a finite number of
variables and that there exist k-universal graphs with no k-extendible induced
subgraphs
