Abstract 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 ≥ 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 ≥ 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.