Math 580
Combinatorial Mathematics, Fall 2012
Instructor: Jozsef Balogh
Office: 233B Illini Hall
Phone: (217) 2441918 (office)
Email: jobal@math.uiuc.edu
Time and place: 12:00 12:50pm MWF, 447 Altgeld Hall
Problem Session: Monday 35:00pm 145 Altgeld Hall
Office Hour: by appointment, designated time: Wednesday, Friday 2:002:50pm
Make up: Wednesday 35pm 145 AH
Midterm exam: October 24 Thursday 6:308:00 pm. 1027 Lincoln Hall
Final exam: December 18: 7:0010:00 PM, Tuesday.
Make up class: Thursday 1011am 447 AH
Advice for Homework:
Homework is usually due on Wednesdays, with solutions distributed on Fridays.
Start early on homework assignments, and ask questions if anything is unclear
or if you need guidance. I can provide a push in the right direction when
people get stuck. The purpose of the homework is to help students understand
the ideas, so it is okay to toss around some thoughts with others. Be sure to
write up solutions on your own. Solutions should include full
justifications in sentences.
The text and class cover classical examples, and working other problems for
homework leads to deeper understanding. I try to cover by Friday almost
all of what is needed to do the homework for the next week; sometimes there
will be an item or two left for Monday.
I hold voluntary collaborative study sessions on Monday 3:004:50pm. Students discuss the homework in small groups; I listen and answer
questions. Whether you attend or not, you should
still try to solve the problems before Monday. The Monday discussion is more
beneficial if you have already tried the problems.
Homework allows you to choose five out of six problems to write up,.
Problems are worth 5 points each, so the maximum score/homework is 25 points.
The test problems are chosen among the recommended problems.
 Homework 1: DUE: Sept 7 (Friday)
WARM UP PROBLEMS: Section 1.1: 3, 4, 5, 7, 8, 11, 12, 14; Section 1.2: 1, 2, 4, 6, 8, 10;
Section 1.3: 1, 3, 4, 5. Do not write up. These problems are easier or shorter to check understanding.
EXTRA PROBLEMS: Section 1.1: 19, 22, 26, 27, 29, 32, 34, 37; Section 1.2: 11, 13, 14, 25, 26, 28, 33, 35, 38, 41, 42, 45, 48, 49, 52.
WRITTEN PROBBLEMS: Do five of the six problems. Type your solution. Prove your statements.
Section 1.1: 34; 37; Section 1.2: 16 (a),(b),(d), 19; Section 1.3: 16, 27;

Homework 2:
DUE: Sept 12 (Wednesday)

Homework 3:
DUE: Sept 19 (Wednesday) From Problem 2 submit only part (b)!

Homework 4:
DUE: Sept 26 (Wednesday)

Homework 5:
DUE: Oct 5 (Friday)

Homework 6:
DUE: Oct 15 (Monday) [NOT WEDNESDAY!!!!!]
WARM UP PROBLEMS: Section 4.2: 2, 3, 5. Section 4.3: 3, 8.
EXTRA PROBLEMS: Section 4.2: 8, 10, 15, 21, 25. Section 4.3: 9, 15.
WRITTEN PROBBLEMS: Section 4.2: (do both) 16, 21. and several others...

Homework 7:
DUE: Oct 29 (Monday)
WARM UP PROBLEMS:
Section 6.1: 2,3,4,6,7,8;
Section 6.2: 1,3;
Section 7.1: 1,2,3,4
EXTRA PROBLEMS:
Section 6.1: 11, 13, 16,17, 20, 25, 29, 30, 31, 35, 38, 39;
Section 6.2: 4,5, 11, 12, 16, 22, 24, 31, 35;
Section 7.1: 16, 17, 18, 20, 21, 29, 31, 33.
WRITTEN PROBBLEMS:
Section 6.1: 21, 24, 30;
Section 6.2: 18, 37;
Section 7.1: 25;

Homework 8:
DUE: Nov 5 (Monday)
WARM UP PROBLEMS:
Section 7.2: 2,4, 6, 7, 8, 10, 14;
Section 7.3: 3,4, 5
EXTRA PROBLEMS:
Section 7.2: 21, 22, 25, 29, 30, 36, 38;
Section 7.3: 7, 9, 14, 17, 21, 26, 29, 30, 31, 43.
WRITTEN PROBBLEMS:
Section 7.1: 21;
Section 7.2: 37, 47
Section 7.3: 6, 29;
Section 8.1: 24

Homework 9:
DUE: Nov 9 (Friday)
WARM UP PROBLEMS:
Section 8.1: 1, 4, 5, 6, 7;
Section 8.2: 1, 3, 6, 7;
Section 8.3: 3, 4, 6, 7;
EXTRA PROBLEMS:
Section 8.1: 11, 15, 20, 26, 31, 33, 37, 41;
Section 8.2: 12, 14, 15, 21, 25, 28, 39, 40, 43;
Section 8.3: 10, 13, 18, 25, 34, 42, 50;
WRITTEN PROBBLEMS:
Section 8.1: 44; TYPO IN part (b), the "at most" should be "at least"
Section 8.2: 21, (27,28 as one problem), 39
Section 8.3: 22, 24;

Homework 10:
DUE: Nov 16 (Friday)

Homework 11:
DUE: Nov 28 (Wednesday)

Homework 12:
DUE: Dec 17 (Monday) noon, via email, strict deadline, no late hw is allowed!
Do 5 problems!
Do at least 2 out of the next 5 problems:
Problem 1. Prove that every balanced partition of the half graph into 2k classes contains at least
k/100 not epsilonregular pairs, where every partition class is fully contained in one of the classes of the halfgraph. Note: The half graph has n vertices in each classes.
Note that k is much larger than 1/epsilon, and n is much larger than k.
Chapter 14.2: 5, 9; 14.3: 14, Note there is a typo in part (b), it should be R(k,k), 22.
Do at least one from Chapter 12: 12.1.15; 12.2.37
Do at least one from Chapter 13: 13.1.9; 13.2.6.
Advice for students in Math 580 / CS 571:
General:
I hope you will enjoy the variety and beauty of the combinatorial problems
discussed in this course. The course provides a fairly thorough study of basic
combinatorial techniques at the graduate level, plus an introduction to some
additional topics.
Background:
The course is an introduction, but at the graduate level. It assumes
``mathematical maturity'', which means familiarity with elementary undergraduate
mathematics and a level of comfort with reading and writing proofs. There is
no specific prerequisite in combinatorics; I will develop all the combinatorics
from the beginning. However, {\it the course moves quickly and covers a lot}.
Some students will have had previous exposure to enumeration or graph theory.
What matters more than specific background is interest in the material and a
commitment to work hard.
Feedback:
Please feel free to offer feedback by email or in office hour about the
lectures, text, topics, homework, etc. Also feel free to ask questions
in class. I will use email to make announcements about the course (including
clarification of items from class), and email is often the easiest and quickest
way to reach me. When I hear about items that are unclear or in error
(including on homeworks), I can send everyone email to fix it.
Homework Solutions:
You may be sure, dear Crito, that inaccurate language is not only in
itself a mistake: it implants evil in men's souls.  Socrates, in Plato's Phaedo.
From past experience, I know that some(?) (ALL!) of you will need to improve your
writing. An important part of doing mathematics is communicating it clearly.
I suggest the following. Instead of solving all the problems first and then
writing them up, write up problems right away when you solve them. You then
have a chance to forget what you wrote for the first few problems while you
work on the others. At the end, before submitting the homework, read what you
wrote earlier. If you find it hard to follow, then graders will find it even
harder! At this point revise your exposition so that it really say what you
had in mind.
When you think of a proof and write it down, it seems clear to you because
the ideas are fresh in your mind. Revising after you have a chance to forget
the thoughts is very important and is the only way you can be confident that
what you wrote expresses what you meant. This process also helps you discover
habits of expression that make your exposition unclear, and then your early
drafts of future work will be better. Writing as you solve gives some benefits
of the revision process without taking too much extra time and without
requiring you to solve all the problems by Saturday. Some of you like professor West's
opinion on correct grammar and usage in mathematical writing are posted at
https://faculty.math.illinois.edu/\~west/grammar.html.
There is much of beauty in combinatorics, so there is much I want to tell you.
The lectures aim to increase your interest in these topics and to convey a
basic understanding of the techniques and proofs and how they fit together.
Sometimes you may not fully understand all details just by attending lecture;
that's why the details are in the text. The lectures are a guide to the text.
Using your notes to indicate the important points, review the text to
understand the details securely (also ask questions if something is not
adequately explained!) Mastery of the material comes from reflecting on it at
your own pace and from solving problems, which is why the homework is so
important.
If you are struggling with the material, I will be happy to give further
explanations. Don't give up without at least trying to get some help or
clarification. I hope you all have a great time and learn a lot!