18.315 Combinatorial Theory (Fall 2005)
Time and place: MWF 10, room 12-142.
Lecturer: Igor Pak, 2-390 (OH on M 11-12).
Course work: weekly home assignments (no final exam).
- 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
- 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
- my new graph-theoretic proof of Dehn's theorem
- Barvinok's algorithm for counting integer points in
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.
R. Diestel, Graph Theory (Graduate Texts in Mathematics),
Springer, 1997 (available electronically
J. Matousek, Lectures on Discrete Geometry (Graduate Texts in Mathematics),
1. See the .ps file or
.pdf file. Click
here for the extensive
literature on problem 6.
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
4. See the .ps file or
5. See the .ps file or
.pdf file. Whitney's article
"A Theorem on Graphs'' is available from
6. See the .ps file or
7. See the .ps file or
8. See the .ps file or
(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.
to return to Igor Pak Home Page.
To e-mail me click
here and delete .zzz
Put 18.315 in the Subject line.
Last updated 9/20/2005