**Who can participate?**The list of research topics and professors proposing them is posted below. In addition to the problem description, each professor gives introductory problems for interested students to solve under "Qualifying problems". If you are considering participating, submit your solutions to the qualifying problems of the research problem that you would like to work on**by August 27th**, directly to the professor whose problem you are interested in.

Enrollment will be based on solutions submitted to qualifying problems to professors and your transcript and reference letter submitted as your application materials to the American office. Decisions on admittance to the research group will be made**by September 3rd**.**Which topics will actually be offered ("stay alive")?**Of the initially offered research problems those will be offered eventually, for which a group of students sign up and are admitted to the program. Ideally, a group would consist of 3-5 students.**Course work: weekly meetings & presentation.**The research groups will meet at least 2 times weekly, two hours each. (At least) one meeting is devoted to group work, when you discuss the problem and possible solutions with your student group, without the professor. The other meetings of the week are spent with your professor who will monitor your group's progress. Whether you meet once or twice/with your Professor on a given week will be decided case-by-case, depending on progress.

__Week 3 - Milestone 1:__The first 2.5 weeks of the course are spent discussing, gathering and studying necessary background information for your problem. By the end of the 3rd week, each student wishing to participate in a research group has to write a summary of the status of the research project at that point — on their own. The summary should consist of stating the problem/aims of the research group, plans on tackling the problem, as well as an outline of work done during the first 2.5 weeks with a write up of results (if any) achieved by that time.

In addition, each research group will have to present the problem they are working on in a 20-minute talk, at a "mini workshop" organized for all research participants, their professors and everyone else interested.At this point professors evaluate progress and final enrollment decisions are made, based on the written summary, oral presentation and work done during the first 2.5 weeks.

Please, note that some research groups may die out or be discontinued after the 3rd week, so plan accordingly. Also, the research class is the oly class where a student wishing to take the course may not be able due, since it is at the discretion of the professor to let students become members of their research group.

__Week 7 - Milestone 2:__at week 7 (just like at week 3), each student is required to submit a (relatively short) report on the work in progress, to their professor. (Thus the report should include a eg description of the problem, as well as the methods used in tackling the it and a write-up of results, if any.)Based on all work up to that point and the written report, students receive "midterm evaluation grades".

__Week 13 - Milestone 3:__: Work continues throughout the semester. At the 13th week each group should present their results at a "Preliminary report session" organized for all RES participants, their professors and everyone else interested.Grading is done on an A-F scale.

Write up of results is continuous and oftentimes streches to after the semester is over.**Course work: writing a paper.**Depending on results obtained all work will be summarized in a paper/research report.- determine/estimate the maximum number
*ex_p(n,F)*of edges in a planar graph*G*of*n*vertices not containing*F*as a subgraph. - determine/estimate the maximum number
*f(n,H)*of copies of*H*in a planar graph*G*of*n*vertices - D. Ghosh, E. Győri, R. R. Martin, A. Paulos, N.Salia, C. Xiao, O. Zamora, The Maximum Number of Paths of Length Four in a Planar Graph, arXiv:2004.09207
- D. Ghosh, E. Győri, R. R. Martin, A. Paulos, C. Xiao, Planar Turán number of the 6-cycle, arXiv:2004.14094

This course is designed in the style of
the Hungarian "TDK" system, allowing *advanced* undergraduates to
become acquainted with research methods in detail and
acquire additional knowledge beyond their obligatory curriculum.
(For a brief English description of the TDK system
see a relevant ELTE University
homepage.)

In this course, a student can choose from the topics/problems listed below and work with
other students and the professor to solve the given problem. All work is
summarized in two individual reports (as explained below) and
ideally a research paper, however that is not expected to achieve given the time constraints.
In addition, during the semester there will be opportunities
to present your work as well.

Participating in the research course may contribute to the successful beginning of a scientific career:
depending on level,
the results obtained can be presented at school, statewide or national
undergraduate meetings ranging from a local Undergradute Seminar at your
home school to
MAA's MathFest.
Papers may also be published in undergraduate research journals
such as
The Rose-Hulman Undergraduate Mathematics Journal,
Involve and
several others.

In some PhD programs, fruitful undergraduate
reserach activity is a prerequisite for admission.

Student research is supervised by professors. Research topics are offered
by them, but students can also propose topics of their own interest.

You can view
articles that were written under the auspices
of the BSM program

**Description:**
The subject is a new, fast developing area of extremal graph theory. There are two types
of basic problems:

The various constructions of extremal graphs make the subject particularly interesting. An interesting
direction is the combination of these basic type problems: what is the maximum number of copies of
*H* in an *n* vertex planar graph not containing *F* as a subgraph.

The starting point of this subject was the classical result that the maximum number of edges in a
planar graph of *n* vertices is *3n-6* if *n> 3*. Many years later, Dowden proved that the maximum
number of edges in a planar graph not containing any 4-cycle is at most *12(n-2)/7* and it is sharp for
infinitely many values of *n*. (For details, see C. Dowden, Extremal C_4-free/C_5-free planar graphs, J.
Graph Theory 83 (2016), 213– 230.) We plan to consider problems of this type when the graphs F and
H are small.

**Description:** Click here.

**Prerequisites:**We shall mostly use graph theoretic and combinatorial methods but
geometric intuition is also useful. Since some of the
rigidity properties are defined by the ranks of certain
matrices, linear algebra tools may also be needed.

**Professor:**Tibor Jordan

**Contact:**tibor.jordan@ttk.elte.hu

**Qualifying problems:**Solve at least four out of the
five exercises at the end of this document.