Zarankiewicz's problem for semi-algebraic hypergraphs

Zarankiewicz's problem asks for the largest possible number of edges in a graph that does not contain a Ku,u subgraph for a fixed positive integer u. Recently, Fox, Pach, Sheffer, Sulk and Zahl [12] considered this problem for semi-algebraic graphs, where vertices are points in Rd and edges are defined by some semi-algebraic relations. In this paper, we extend this idea to semi-algebraic hypergraphs. For each k≥2, we find an upper bound on the number of hyperedges in a k-uniform k-partite semi-algebraic hypergraph without Ku1,…,uk for fixed positive integers u1,…,uk. When k=2, this bound matches the one of Fox et al. and when k=3, it isO((mnp)2d2d+1+ε+m(np)dd+1+ε+n(mp)dd+1+ε+p(mn)dd+1+ε+mn+np+pm), where m,n,p are the sizes of the parts of the tripartite hypergraph and ε is an arbitrarily small positive constant. We then present applications of this result to a variant of the unit area problem, the unit minor problem and intersection hypergraphs.