Embedding maximal cliques of sets in maximal cliques of bigger sets

Abstract Characterizations are obtained of the maximal ( k + s )-cliques that contain a given maximal k -clique as a substructure: (1) when s = 1; (2) for arbitrary s when no line of the clique contains exactly one point of the subclique. These characterizations are used to obtain maximal cliques which have fewer lines (for given k ) than previously known examples.