Submission date: 27 September 2017

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 ∀ ∃-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.

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Full text arXiv 1709.09626: pdf, ps.