We study a query model of computation in which an n-vertex k-hypergraph can
be accessed only via its independence oracle or via its colourful independence
oracle, and each oracle query may incur a cost depending on the size of the
query. In each of these models, we obtain oracle algorithms to approximately
count the hypergraph's edges, and we unconditionally prove that no oracle
algorithm for this problem can have significantly smaller worst-case oracle
cost than our algorithms