In  his famous paper of 1935, Phillip Hall answered the following question:

What are necessary and sufficient conditions on a collection of finite sets S1, S2, . . . , Sn such that there exists an n-tuple (x1, x2, . . . , xn) of n distinct elements such that xi is in Si?

We call such an n-tuple a system of distinct representatives.  Since this result, there have been many related results, applications and generalizations of Hall’s Theorem. 

In this presentation we will state a more general question to that above and consider how Hall’s Theorem may apply.  We will discuss how previous work may be viewed through this generalization and present a few applications of this theory to universal algebra and graph coloring problems.