  We study intrinsically linked graphs where we require that every embedding of
the graph contains not just a non-split link, but a link that satisfies some
additional property. Examples of properties we address in this paper are: a two
component link with lk(A,L) = k2^r, k not 0, a non-split n-component link where
all linking numbers are even, or an n-component link with components L, A_i
where lk(L,A_i) = 3k, k not 0. Links with other properties are considered as
well. For a given property, we prove that every embedding of a certain complete
graph contains a link with that property. The size of the complete graph is
determined by the property in question.
