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Abstract A bipartite graph $$H = \left (V_1, V_2; E \right )$$ with $$\lvert V_1\rvert + \lvert V_2\rvert = n$$ is semilinear if $$V_i \subseteq \mathbb {R}^{d_i}$$ for some $$d_i$$ and the edge relation E consists of the pairs of points $$(x_1, x_2) \in V_1 \times V_2$$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $$d_1 + d_2$$ variables for some s . We show that for a fixed k , the number of edges in a $$K_{k,k}$$ -free semilinear H is almost linear in n , namely $$\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ for any $$\varepsilon> 0$$ ; and more generally, $$\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$$ for a $$K_{k, \dotsc ,k}$$ -free semilinear r -partite r -uniform hypergraph. As an application, we obtain the following incidence bound: given $$n_1$$ points and $$n_2$$ open boxes with axis-parallel sides in $$\mathbb {R}^d$$ such that their incidence graph is $$K_{k,k}$$ -free, there can be at most $$O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o -minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).