  We study the theory $T_{m,n}$ of existentially closed incidence structures
omitting the complete incidence structure $K_{m,n}$, which can also be viewed
as existentially closed $K_{m,n}$-free bipartite graphs. In the case $m = n =
2$, this is the theory of existentially closed projective planes. We give an
$\forall\exists$-axiomatization of $T_{m,n}$, show that $T_{m,n}$ does not have
a countable saturated model when $m,n\geq 2$, and show that the existence of a
prime model for $T_{2,2}$ is equivalent to a longstanding open question about
finite projective planes. Finally, we analyze model theoretic notions of
complexity for $T_{m,n}$. We show that $T_{m,n}$ is NSOP$_1$, but not simple
when $m,n\geq 2$, and we show that $T_{m,n}$ has weak elimination of
imaginaries but not full elimination of imaginaries. These results rely on
combinatorial characterizations of various notions of independence, including
algebraic independence, Kim independence, and forking independence.
