**Content:**

- Extremal graph theory
- Turan theorem (extremal graphs with no k-cliques)
- graph with large degree and girth
- Posa theorem, long cycles in graphs
- various extremal results on graph colorings

- Traditional graph theory
- Hamiltonicity (Dirac, Fleischner theorems)
- 5-color theorem, Brooks theorem, other results on graph colorings
- Menger theorem
- Tutte polynomial

- Set combinatorics
- set colorings (Erdos-Lovasz)
- chains and antichains in posets (Dilworth, Sperner)
- spaces of polynomials (Frankl-Wilson, Bollobas)
- colorings of R^d, disproof of the Borsuk conjecture

- Additive combinatorics
- Schur theorem
- van der Waerden theorem

- Enumerative combinatorics:
- non-intersecting paths lemma
- MacMahon's formula via a series of bijections
- Fomin's extension and probabilistic applications
- heaps of pieces, matrix tree theorem from here
- from spanning trees to domino tilings
- height functions and local move connectivity of domino tilings
- Kastelyn determinants, the number of domino tilings

- Geometric combinatorics:
- Helly theorem and Borsuk theorem in R^2.
- Pogorelov's double counting proof of the Cauchy theorem
- Dehn's original proof of the rigidity theorem + enumerative applications
- my new graph-theoretic proof of Dehn's theorem
- Barvinok's algorithm for counting integer points in polytopes

**Textbooks:**

R. P. Stanley, *Enumerative Combinatorics*, vol. I, II,
Cambridge University Press, 1999.

B. Bollobas, *Modern Graph Theory*
(Graduate Texts in Mathematics), Springer, 1998.

B. Bollobas, *Extremal Graph Theory*, Dover, New York, 2004.

S. Jukna, *Extremal Combinatorics*, Springer, Berlin, 2000.

**Additional Reading:**

R. Diestel, *Graph Theory* (Graduate Texts in Mathematics),
Springer, 1997 (available electronically
here).

J. Matousek, *Lectures on Discrete Geometry* (Graduate Texts in Mathematics),
Springer, 2002.

**2.** See the .ps file or
.pdf file. Try to either view
the homework on a computer screen or print it on a color printer.

**3.** See the .ps file or
.pdf file.

**4.** See the .ps file or
.pdf file.

**5.** See the .ps file or
.pdf file. Whitney's article
"*A Theorem on Graphs*'' is available from
JSTOR or
here.

**6.** See the .ps file or
.pdf file.

**7.** See the .ps file or
.pdf file.

**8.** See the .ps file or
.pdf file
(note a different due date!)

**Collaboration policy:** The collaboration is encouraged
with a few simple rules. On every problem not more than four people
can collaborate. Every student writes her/his own solution.
For each problem, all collaborators should be listed.

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*Last updated 9/20/2005*