Eric Li and Anthony Thompson -- Hall's Marriage Theorem

   This is a pretty good exposition of the ``Marriage Theorem'' and some of its applications.
   You have a lot of different interesting and representative examples.  However:

   (a) I think you have learned that writing about mathematics is different (and harder) 
   than other sorts of writing because you have to be so careful to state things precisely.
   Unfortunately, this means that there are quite a few things in the paper that 
   are just not correct as stated.  
 
   (b) As mathematical writing, your paper also has a defect at the start.  Namely,
   you introduce a topic and use its terminology in an example where you haven't
   defined the terms or laid other necessary groundwork yet.  For instance, you 
   talk in some detail about examples from Bryant's book at the start (pages 1 and 2)
   without describing what the sets of boys are.  It's difficult (if not impossible)
   to follow the examples and understand what they are supposed to illustrate without
   knowing the details. 

   (c) I think you may also have written this ``by accretion'' (i.e. only by 
   adding to what you had done before and not revising and reworking things when
   you decided you wanted to include something new).  This means that the organzation
   of the paper is not very good.  It just meanders without much of a progression
   for the basics to the more complicated examples.  Also, the transversal (SDR) 
   form of the theorem is only described very late.  To me, at least, it's the most 
   basic form and it's how you really should think about applying the theorem in 
   most cases.  In other words the ``Marriage Theorem'' is not about
   matching boys and girls and arranging marriages.  It's really about finding 
   SDRs in a collection of finite sets.  The ``marriages'' are only an *analogy*
   that help you understand the more abstract version of the theorem.  And then
   that abstract version can be used in lots of different situations.

   Some specific comments:

   1)  Bottom of page 5:  "It is also possible for a perfect matching to not exist even 
       when the bipartite theorem and Hall’s marriage theorem hold true. This can happen 
       when a constricted set exist, where the edges affect the result of a perfect matching."

       > This is not clear for several reasons.  First, the wording 
         "bipartite theorem" does not make sense.  The bipartite property is a 
         a property that a graph can either have or not; it's not a theorem. I 
         think what you mean here is that in order for a *perfect* matching to exist 
         (one where there is exactly one edge in the matching incident with each 
         vertex), then it must be true that |V_1| = |V_2| so the 
         total number of vertices is even.  If you don't require that the matching 
         is perfect in that sense, then there is a matching in the bipartite graph
         (V_1,E,V_2) IF AND ONLY IF the Hall condition |U_{i in I} A_i| >= |I| 
         for all subsets I of the indices of the vertices in V_1, and 
         where A_i is the set of vertices in V_2 connected to vertex i in V_1. 
         But then the matching is just a collection of edges joining each of the 
         vertices in V_1 to distinct vertices in V_2 and some of the vertices in
         V_2 might not be "used."     

   2)  page 6: The |U_{i in I} A_i| >= |I| *is* the Hall condition
       
   3)  page 6:  "The essentially means that the subset of |I| has a number of sets less 
       than or equal to the distinct elements in the union over the subset of I."

       > Not exactly!  This is an example of what I meant above about the need to 
       say things precisely when writing about mathematics.  The Hall condition says that
       the union of any collection of the A_i, indexed by a subset I of [n], contains 
       at least as many elements as the number of elements in I (the number of sets
       in the collection).  

   4)  page 6:  A1. "It is important to note that within the transversal
       those values could go in a different order they are not set."

       > Not exactly!  There is an ordering on the collection (A_1, A_2, ... , A_n)
       given by the ordering on the numbers 1,2, ... , n.  The elements of the 
       SDR (transversal) (a_1,a_2, ... , a_n) have to be ordered in the same
       way because a_i is supposed to be an element of the set A_i for every i. 

   5)  page 7:  "So, the number of ‘yes’ swipes by men are
       represented by A_i and |I| is represented by the number of women on Tinder.
      
       > No.  Each man using Tinder has a set of women he has swiped 'yes' on. 
       If the set of men using Tinder is labeled with the integers in [n], then
       for each i, the set of women man i has swiped `yes' on is A_i.  
       |A_i| is the number of yes swipes man i has made.  If I is a subset of [n], |I| is 
       the number of men in some collection of men, not the number of women.  

   6)  page 7:  "This is true because if there is a single element
       in each set then a transversal cannot be created."

       > Not exactly!  It depends on what the elements of the sets are.
       If they are all distinct, then there is a transversal.
  
   7)  page 11: In order to create a 4 x 5 Latin Rectangle, one can find a 
       system of distinct representatives of the remaining numbers not
       in the set.

       > This is not exactly right!  You are talking about the family 
       of sets (A_1,A_2,..., A_n) where for each j, 1 <= j <= n, A_j is the 
       remaining numbers not used *in column j.*  The transversal here
       represents a row that can be adjoined to the bottom of the matrix
       to create a Latin rectangle with one more row.  


   Final Project Grade Computation 
 
   Bibliography:  10/10

   Paper:         52/60

   Presentation:  30/30

   Total:        92/100