  Parent-identifying schemes provide a way to identify causes from effects for
some information systems such as digital fingerprinting and group testing. In
this paper, we consider combinatorial structures for parent-identifying
schemes. First, we establish an equivalent relationship between
parent-identifying schemes and forbidden configurations. Based on this
relationship, we derive probabilistic existence lower bounds for two related
combinatorial structures, that is, $t$-parent-identifying set systems
($t$-IPPS) and $t$-multimedia parent-identifying codes ($t$-MIPPC), which are
used in broadcast encryption and multimedia fingerprinting respectively. The
probabilistic lower bound for the maximum size of a $t$-IPPS has the
asymptotically optimal order of magnitude in many cases, and that for $t$-MIPPC
provides the asymptotically optimal code rate when $t=2$ and the best known
asymptotic code rate when $t\geq 3$. Furthermore, we analyze the structure of
$2$-IPPS and prove some bounds for certain cases.
