
Contact Information
233B Illini Hall, MC-382
1409 W. Green Street
Urbana, IL 61801
Research Areas
Research Interests
graph theory, extremal combinatorics, additive combinatorics
Research Description
Balogh’s main work is the development of the container theorem and finding several surprising applications of the method. Balogh, Morris, and Samotij published their work in the article, “Independent Sets in Hypergraphs,” in Journal of the American Mathematical Society 28 (2015). With a suitably defined notion of containment, the container theorem states that there exists a small collection of vertex subsets (containers) in an r-uniform hypergraph such that every independent set of the hypergraph is contained within a subset in the collection, with no subset in the collection inducing too many edges. This method is an effective tool to obtain probabilistic and counting variants of several classical results in graph theory. It helped to resolve or made breakthrough of several old conjectures (among other’s Erdös’), such as counting combinatorial objects, like maximal triangle-free graphs, maximal sum-free sets; union-free sets; it has applications in Ramsey theory, discrete geometry, theoretical computer science. The container method has become an important tool for obtaining additional results in these and other areas, there are over 40 papers using the method as one of their main tools.
Education
PhD Mathematics, University of Memphis, 2001
External Links
Honors & Awards
NSF CAREER Award, Methods and Outreach in Modern Combinatorics" (2008–2013)
Simons Fellow (2013–2014)
Marie Curie Fellow (2013–2017)
George Pólya Prize in Combinatorics, SIAM, 2016
Recent Publications
Balogh, J., Krueger, R. A., & Luo, H. (2023). Sharp threshold for the Erdős–Ko–Rado theorem. Random Structures and Algorithms, 62(1), 3-28. https://doi.org/10.1002/rsa.21090
Akhmejanova, M., Balogh, J., & Shabanov, D. (2022). Chain method for panchromatic colorings of hypergraphs. Discrete Applied Mathematics, 321, 72-81. https://doi.org/10.1016/j.dam.2022.06.005
Araujo, I., Balogh, J., & Wang, Y. (2022). Maximum determinant and permanent of sparse 0-1 matrices. Linear Algebra and Its Applications, 645, 194-228. https://doi.org/10.1016/j.laa.2022.03.020
Balog, J., Kostochka, A., Lavrov, M., & Liu, X. (2022). Monochromatic paths and cycles in 2-edge-coloured graphs with large minimum degree. Combinatorics Probability and Computing, 31(1), 109-122. https://doi.org/10.1017/S0963548321000201
Balogh, J., Clemen, F. C., Heath, E., & Lavrov, M. (2022). A strengthening of the Erdős–Szekeres Theorem. European Journal of Combinatorics, 101, [103456]. https://doi.org/10.1016/j.ejc.2021.103456