18.315 Combinatorial Theory (Fall 2005)

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.

Home assignments:

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.

Click here 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