Problem sets will be due about once every two weeks. You will be
asked to hand in a subset of your choosing of specified size from a
list of problems. Hand in **at most
one** part from any multipart problem. Each problem has
a difficulty factor [d], such as [3-]. This is converted into a weight
w(d), as follows:

- w(1-) = 0.1, w(1) = 0.3, w(1+) = 0.5
- w(2-) = 1, w(2) = 2, w(2+) = 3
- w(3-) = 5, w(3) = 8, w(3+) = 12
- w(4-) = 18, w(4) = 25, w(4+) = 50
- w(5-), w(5): depends on solution

Problems should be solved primarily on your own. Some "reasonable"
collaboration is permitted, but you shouldn't just obtain the solution
from another source. **Do not** hand in a solution that
you did not obtain on your own or did not obtain by collaboration with
another student in the course! You should also not hand in a problem
that you already know how to solve. Also, hand in **at most one
part** of a multipart problem (such as 3.42, which has two parts).

Problem sets can be turned in during class or sent by email to the
grader (gaetz@mit.edu) by the beginning of class on the due date.

Problems come from the following sources:

- Chapter 3 exercises
- Additional problems (possibly not yet complete)

**Problem Set 1:** EC1, Chapter 3, problems 14,
31(a), 34, 39, 42, 44, 46; additional A1, A2, A3, A4. (A problem
number preceded by "A" is an additional
problem.) Problem 3.14(a) will be given the weight 4, since it
seems somewhat harder than a typical 2+ problem. Problem 3.14(b) is
for those familiar with the terminology or willing to look it up. In
regard to Problem 3.39, the definition of finitary distributive
lattice in the text is incorrect. See
the errata
(item for page 253) for the correct definition. Hand in your "best"
five problems by the end of class on Monday, September 18.

**Problem Set 2:** EC1, Chapter 3, problems 45(a),
52, 53, 54, 57, 58, 60(a,c); additional A5, A6. Hand in your best four
problems by the end of class on Monday, October 2.

**Problem Set 3:** EC1, Chapter 3, problems 85, 89,
90, 91, 98, 99, 100(d), 114; additional A10. Hand in your best four
problems by the end of class on Monday, October 16. No further
problems are forthcoming. **Note
#1.** The difficulty rating [3+] of 91(b) assumes
that you solve the problem from scratch. If you do it using certain
deep outside results, the rating goes down to
[3-]. **Note #2.** At the
moment the only solution I know for 100(d) uses some results that
have not been covered in class and which a "typical" graduate
student wouldn't know. I seem to remember that I once knew a simpler
solution. For the time being the difficulty rating is raised to [3],
though perhaps this will be downgraded again to [3-] if someone
comes up with a simple solution. **Note
#3.** For 3.98 and 3.114(a), note the
corrections
here.

**Problem Set 4:** EC1, Chapter 3, problems
62(a-e), 63, 64(a,b), 127(a-c), 128, 129, 134, 144, 148, 167(a),
168; additional A8, A11. No further problems are forthcoming. Hand
in your best four problems by the end of class on Friday, October
27. **Hint for A11.**
The crucial property of *U _{k}* is that every zero of the

**Problem Set 5:** EC1, Chapter 3, problems 3.178,
3.183(e), 3.185(f-i); additional A7, A12, A13, A14, A16,
A17. No further problems are forthcoming. Hand
in your best four problems by the end of class on Wednesday,
November 8. Problem A7(a) has a weight of 4 (between [2+] and [3-] in
difficulty). A refinement of Example 3.18.10 may prove useful for A7(a).
Problem A13(b) is rated [3]. However, perhaps there is a simpler
solution than the one that I
know. **Note.** Problem A16(c)
is trivial as stated, so it has been removed.

**Problem Set 6:** EC1, Chapter 3, problems 3.189,
3.191; additional A9, A18, A19, A20, A21, A22, A23, A24. No
further problems are forthcoming. Hand in your best four problems
by the end of class on Monday, November 20. However, do at most
one of A19 and
A20. **Note.** Problem A19
is not so bad using a difficult exercise in Chapter 3. Is there a
simpler solution avoiding this exercise? It is okay to use the
result of A20 in a solution to
A19. **Note.** Problem A9
was originally stated incorrectly. It was corrected on November
8. **Note.** Problem A16(c)
(part of Problem Set 5) is trivial as stated, so it has been
removed. **Note.** It seems
to me that Exercise 3.189 might be much weaker than
possible. Possibly for any *d*≥2, there is a unique (up
to isomorphism) Eulerian poset *P* of rank *d*+1
and with *d* atoms, such that *P*-{1} is
simplicial. Can anyone prove this? It might not be so difficult.
**Update.** Actually, it
is false. Exercise 3.189 seems reasonable, but the difficulty
rating [2+] seems too low to me (unless I have forgotten a simpler
argument), so it now is rated [3-]. The result of this exercise
is also true when *d* is odd, but the proof is much
easier. **Updated
update.** Christian Gaetz found a [2+] level
solution to 3.189 which is probably what I had in mind when I
chose the [2+] rating, so the rating is now downgraded back to
[2+]. As a continuation of 3.189, I have added A23, with a weight
of 4 (difficulty rating between [2+] and [3-]). Do at most one of
3.189 and A23.

**Problem Set 7:** EC1, Chapter 3, problems
3.211, 3.212, 3.215(b); additional A25, A26, A27. Hand in your
best four problems by Monday, December
4. No further problems are forthcoming. Sadly, this is
the final required pset. Be sure to take a look at
the EC1
errata before working on 3.211.

**Self problems:** due by the end of class on
Monday, December 11. At the
course Home Page the following
is stated: you should hand in two problems of your own
with difficulty ratings and solutions. They should be
related to partially ordered sets. I certainly don't
expect them to be publishable in general. Grading of
these "self problems" will be based more on elegance,
originality, and pedagogical value than on
difficulty. The self problems count 20% of the course
grade.

**Problem Set 8:** hand in all problems from
Chapter 3 not previously assigned, without looking at the
solutions. No due date, and no affect on the course grade, but a
good way to learn more about posets.