I enjoy doing research with undergradues. If you are an undergradute student and are interested to work on combinatorial problems, feel free to reach out to me.
Also, I recommend taking a look at the Illinois Geometry Lab (IGL). The Illinois Geometry Lab's mission is to enhance and support undergraduate research within the Department of Mathematics at UIUC and to support departmental efforts to engage local, state and national communities through research. I was a graduate student team leader for two undergraduate research projects, each with 4-5 undergraduate students and a faculty member. Here a description of past undergraduate projects, hosted by the IGL:
Chasing the Threshold Bias of the 3-AP Game (Cao, Li, Tatum, Xoubi, Yin)
Faculty member: S. English
In a Maker-Breaker game there are two players, Maker and Breaker, where Maker wins if they create a specified structure while Breaker wins if they prevent Maker from winning indefinitely. A 3-AP is a sequence of three distinct integers a, b, c such that b-a = c-b. The 3-AP game is a Maker-Breaker game played on [n] where every round Breaker selects q unclaimed integers for every Maker's one integer. Maker is trying to select points such that they have a 3-AP and Breaker is trying to prevent this. The main question of interest is determining the threshold bias q*(n), that is the minimum value of q=q(n) for which Breaker has a winning strategy. Kusch, Rué, Spiegel and Szabó initially asked this question and proved bounds on q*(n). We find new strategies for both Maker and Breaker which improve the existing bounds.
Hypergraph Turán Problems (Liu, Roe, Wang, Zhou)
Faculty member: J. Balogh
What is the maximum number of edges a graph on n vertices can have not containing a triangle? The answer to this question is given by Mantel's theorem, a classical theorem in graph theory. It is part of a problem class called Turán problems, a cornerstone of extremal combinatorics. In general, they ask to maximize a certain (hyper-) graph parameter, such as the number of edges, while satisfying a sparsening condition. Anwering these questions involves constructing extremal examples and proving that they are optimal.
We looked at common proof techniques such as counting arguments, stability arguments and computer-based optimization, and applied them to a subclass of problems which just recently were introduced.