Theory of linear and integer programming
Theory of linear and integer programming
Lenses in arrangements of pseudo-circles and their applications
Proceedings of the eighteenth annual symposium on Computational geometry
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Intersections of Leray complexes and regularity of monomial ideals
Journal of Combinatorial Theory Series A
On s-intersecting curves and related problems
Proceedings of the twenty-fourth annual symposium on Computational geometry
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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.