Ron Aharoni (Technion): Choice functions

There are two types of choice functions, one in which the domain is demanded to be large (the classical case being Hall's theorem, where all men are to be married) and one in which the range is required to be large (a classical case is the Lovasz-Barany colorful Caratheodory theorem, in which the range should contain a given vector in its convex hull). Results in the first are usually Hall-like - if every k sets contain "many" elements, then there exists a choice function as required. Results in the second type are usually of the form "if there are many sets, each being large, then...". We show that this is not a divine decree: there are Hall-like theorems also for the second family.