Given sets *E*_{1}, ..., *E*_{n}, a rainbow set consists of at most *1* element from each *E*_{i}. Bárát, Gyárfas and Sárkozy conjectured that *2n* matchings of size *n* (on a common vertex set) have a rainbow matching of size *n*. The conjecture can be thought as a generalization of Drisko's theorem: *2n*−*1* perfect matchings of size *n* have a rainbow perfect matching. In this talk, we will present a short proof of Drisko's theorem using Bárány's colorful Carathéodory theorem. This new proof leads to the discovery of a fractional version of the conjecture: Let *n* be an integer or a half integer. If the fractional matching number of each of the *2n* graphs is at least *n*, then there is a rainbow edge set of fractional matching number at least *n*.

Joint work with Ron Aharoni and Ron Holzman.

Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220.