Course: MCS 583, Extremal Combinatorics
Call no: 44948
Time: MWF 9:00-9:50pm
Place: 300 Lincoln Hall and Online

Professor: Dhruv Mubayi
Office: 620 SEO
E-mail: mubayi@uic.edu
Course Web Page: http://www.math.uic.edu/~mubayi/583/Fall21/ExtremalFall21.html
Office Hours: W 1-3, on Zoom

Grading Policies:
You grade will be based on occasional homework assignments, class presentations, and discussions.

Face Masks: Masks covering both the mouth and nose must be worn at all times by all students, faculty, and staff while on campus and inside any building regardless of vaccination status. If you do not wear a mask, you will be asked to leave the classroom and will not be allowed back in class unless or until you wear a mask. If you have forgotten your mask, you may pick one up from one of the student information desks on campus during the first two weeks of campus. Students who do not comply with the mask-wearing policy will be reported to the Dean of Students. Eating and drinking are not allowed in classrooms.

Disability Policy: UIC is committed to full inclusion and participation of people with disabilities in all aspects of university life. Students who face or anticipate disability-related barriers while at UIC should connect with the Disability Resource Center (DRC) at drc.uic.edu, drc@uic.edu, or at (312) 413-2183 to create a plan for reasonable accommodations. In order to receive accommodations, students must disclose disability to the DRC, complete an interactive registration process with the DRC, and provide their course instructor with a Letter of Accommodation (LOA). Course instructors in receipt of an LOA will work with the student and the DRC to implement approved accommodations.

Prerequisite: An undergraduate course in combinatorics/graph theory or probability, and the mathematical maturity of a (relatively advanced) graduate student.

Course Description: Extremal combinatorics studies the extreme value of a parameter over a class of discrete objects. The subject has been growing for the past century and by now it encompasses some of the most important contributions to combinatorics and has applications to many other disciplines including discrete geometry, number theory, coding theory, computer science. This course will study the modern developments in the subject focusing on Ramsey theory and extremal graph and hypergraph theory. Throughout the course open problems will be presented that are suitable for thesis research.

A sampling of topics:

• Ramsey numbers of graphs and hypergraphs including recent work on asymmetric Ramsey numbers for hypergraphs, the Erd\H os-Rogers problem, and multicolor Ramsey numbers.
• Methods in extremal graph and set theory, including classical extremal graph theory (Zarankiewicz problem, Erdos-Simonovits-Stone theorem) and more recent probabilistic tools (dependent random choice) and algebraic constructions (Projective Norm graphs); extensions of classical results in extremal set theory (Erdos-Ko-Rado type theorems), the delta system method, the random walk method, and more recent approaches using sampling and shadows.
• Graph and Hypergraph Regularity Lemmas and accompanying removal lemmas; applications to Roth's theorems on arithmetic progressions and the multidimensional Szemer\'edi theorem.
• Quasirandom structures focusing on recent developments on hypergraph quasirandomness.