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Preprint Number 1310
1310. Gabriel Conant and Alex Kruckman Independence in generic incidence structures E-mail: 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. Mathematics Subject Classification: Keywords and phrases: |

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