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Combinatorial Geometry (CS 518)
Course Synopsis:
This is a graduate level course in combinatorial geometry. Both recent and classical
topics in combinatorial geometry will be covered in this course, with emphasis on
currently open research problems. There are two primary texts, which will be
supplemented with recent research papers which will be distributed in class. The students
will be expected to know elementary probability theory, as well as have taken a course in
discrete mathematics. The grading consists of home works, as well as end-of-course
presentations. At the beginning of the course, students will be expected to pick a research
topic related to some open problem in geometry, and read the necessary papers on that
topic, and present them at the end of the course.
Course Outline:
1. Introduction. Euclidean geometry, Common techniques such as probabilistic methods,
comparability graphs, inclusion-exclusion principles, elementary counting ideas like
double-counting, pigeon-hole principle.
2. Geometric Graphs. Planar graphs, bounding number of edges in planar graphs, planar
separators, bounding edges based on intersection and non-intersection, crossing numbers,
bisection width.
3. Points. Repeated angles, large angles, repeated distances, distinct distances, centerpoints, partitioning cuts.
4. Lines. Point-line duality, arrangements, complexity of a collection of cells in
arrangements, cuttings, point-line incidences, Hopcroft's problem.
5. Rectangles. Ramsey theory for rectangles, partitioning rectangles, conflict-free
6. Convex objects. Helly's theorem, Ramsey theory, separators, cutting glass.
7. Geometric set systems. Fractional transversals, range spaces, VC-dimension, eps-nets,
spanning trees with low stabbing numbers.
-Course presentation: 50%
-Homework: 50%
Reading material:
-Primary textbooks:
Combinatorial Geometry, by P. Agarwal and J. Pach.
Discrete and Computational Geometry, by J. Matousek.
Supplemented with recent papers.