This course is designed in the style of the Hungarian "TDK" system, allowing advanced undergraduates to become acquainted with research methods and means 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 a paper and during the semester there will be opportunities to present your work as well.

This 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 or Involve.

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

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


The list of research topics and professors proposing them can be seen below. Contact the professor whose problem you are interested in at the Welcome Party, but read everything carefully below first.


  1. Title: Extremal sets of the vertices of the hypercube (over GDF(2))

    Description: We plan to investigate (cases of) the following general question: How many vertices (maybe of certain further property, like of fixed weight) of the n-dimensional hypercube can be picked such that subspace spanned by them - over GF(2) - does not contain or does not intersect certain configurations of the hypercube (vertices, vertices of given weight, subspaces, hyperplanes, etc.)

    In this project you will understand the structure of the hypercube over the reals and GF(2), develop algebraic methods to solve extremal set theoretical problems and establish constructions and will reach - in the worst case - some concrete results.

    Prerequisites: basic combinatorics and linear algebra
    Best for: students who intend to do research in algebra or combinatorics
    Professor: Dr. Dezsô Miklós
    ASSIGNMENT FOR THE FIRST WEEK: exercises and related reading

  2. Title: Large set in the line without a given pattern

    Description: In the real line by a rectangle we mean points of the form x, x+a, x+b, x+a+b, and by the area of such a rectangle we mean the product ab. The question is how large a subset of the real line can be without having rectangle of area at least 1. One can get a more precise question in a number of different ways depending on the the assumption about the set and the measurement we use, but, as it can be easily seen, they are all equivalent, so students can choose the form they prefer. In the measure theoretic form we consider Lebesgue measurable sets and we use Lebesgue measure. In the combinatorial form the sets are finite unions of intervals and we consider the sum of the lengths of the intervals. In the number theory form the points of the sets are chosen from a grid and we consider the number of chosen points multiplied by the length of the grid. The methods of all the above areas can be used and it is not clear at all which will be useful. The motivation comes from a long standing very hard unsolved problem of A. Carbery. An arbitrarily large set without rectangle of area at least 1 would immediately answer Carbery's problem.

    Prerequisites: none
    Best for: Students with good problem solving skills who would like to work heavily on unsolved problems.
    Professor: Dr. Tamás Keleti
    ASSIGNMENT FOR THE FIRST WEEK: exercises (do at least part (a), if you are interested in participating)

  3. Title: Coloring t-intersecting hypergraphs

    Description: click here for the description of the problem

    Prerequisites: basic combinatorics;
    Best for: students who intend to do research in combinatorics
    Professor: Dr. András Gyárfás
    ASSIGNEMENT FOR THE FIRST WEEK: will be given at the Welcome Party

  4. Title: Realizations of graphical degree sequences

    Description: Let D = d_1, d_2, ... d_n be a series of positive integers. We say that D is a graphical degree sequence, if there exists a simple graph G whose degrees are the given numbers. G is called a realization of the degree sequence. Erdos and Gallai gave sufficient and necessary conditions when a degree sequence is graphical, and Havel and Hakimi gave an algorithm how to construct one realization if exists (otherwise report that the sequence is not graphical).

    We are going to study the properties of the realizations of graphical degree sequences, extending it to mulipartite graphs, restricting the solutions to some subsets of solutions with prescribed properties, etc., based on our earlier results, see: article 1, article 2 and article 3.

    Prerequisites: basic combinatorics
    Best for: students who intend to do research in graph theory
    Professor: Dr. István Miklós
    ASSIGNMENT FOR THE FIRST WEEK: Read Chapter 15 and solve exercises 15.1-15.6.