Transversal numbers for hypergraphs arising in geometry

The (p,q) theorem of Alon and Kleitman asserts that if F is a family of convex sets in R^d satisfying the (p,q) condition for some p>=q>=d+1 (i.e. among any p sets of F, some q have a common point) then the transversal number of F is bounded by a function of d, p, and q. By similar methods, we prove a (p,q) theorem for abstract set systems F. The key assumption is a fractional Helly property for the system F^@? of all intersections of sets in F. We also obtain a topological (p,d+1) theorem (where we assume that F is a good cover in R^d or, more generally, that the nerve of F is d-Leray), as well as a (p,2^d) theorem for convex lattice sets in Z^d. We provide examples illustrating that some of the assumptions cannot be weakened, and an example showing that no (p,q) theorem, even in a weak sense, holds for stabbing of convex sets by lines in R^3.

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