Zarankiewicz’s problem for semilinear hypergraphs

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