The Dynamic Descriptive Complexity of k-Clique

In this work the dynamic descriptive complexity of the k-clique query is studied in a framework introduced by Patnaik and Immerman. 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.