The dynamic descriptive complexity of k-clique

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 quantifierfree 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. (C) 2017 Elsevier Inc. All rights reserved.