  In this work the dynamic descriptive complexity of the k-clique query is
studied. It is shown that when edges may only be inserted then k-clique can be
maintained by a quantifier-free update program of arity k-1, but it cannot be
maintained by a quantifier-free update program of arity k-2 (even in the
presence of unary auxiliary functions). This establishes an arity hierarchy for
graph queries for quantifier-free update programs under insertions. The proof
of the lower bound uses upper and lower bounds for Ramsey numbers.
