Zarankiewicz'S Problem For Semilinear Hypergraphs

A bipartite graph H = (V-1, V-2; E) with vertical bar V1 vertical bar + |V-2 vertical bar = = is semilinear if V-i subset of R-di for some 38 and the edge relation E consists of the pairs of points (x(1), x(2)) is an element of V-1 x 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-k-free semilinear H is almost linear in n, namely vertical bar E vertical bar = O-s,O-k,O-epsilon (n(1+epsilon)) for any epsilon > 0; and more generally, vertical bar E vertical bar = O-s,O-k,O-r,O-epsilon (n(r-1+epsilon)) for a K-k,K-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 R-d such that their incidence graph is K-k,K-k-free, there can be at most O-k,O-epsilon (n(1+epsilon)) 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).