  We consider the problem of counting, in a given graph, the number of induced
k-vertex subgraphs which have an even number of edges, and also the
complementary problem of counting the k-vertex induced subgraphs having an odd
number of edges. We demonstrate that both problems are #W[1]-hard when
parameterised by k, in fact proving a somewhat stronger result about counting
subgraphs with a property that only holds for some subset of k-vertex subgraphs
which have an even (respectively odd) number of edges. On the other hand, we
show that the problems of counting even and odd k-vertex induced subgraphs both
admit an FPTRAS. These approximation schemes are based on a surprising
structural result, which exploits ideas from Ramsey theory.
