
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
Awards and Honors
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
External Links
Recent Publications
Balogh, J., Dudek, A., & Li, L. (2021). An analogue of the Erdős–Gallai theorem for random graphs. European Journal of Combinatorics, 91, [103200]. https://doi.org/10.1016/j.ejc.2020.103200
Balogh, J., Garcia, R. I., & Li, L. (2021). Independent sets in the middle two layers of Boolean lattice. Journal of Combinatorial Theory. Series A, 178, [105341]. https://doi.org/10.1016/j.jcta.2020.105341
Balogh, J., Katona, G. O. H., Linz, W., & Tuza, Z. (2021). The domination number of the graph defined by two levels of the n-cube, II. European Journal of Combinatorics, 91, [103201]. https://doi.org/10.1016/j.ejc.2020.103201
Balogh, J., Petříčková, Š., & Wagner, A. Z. (2020). Families in posets minimizing the number of comparable pairs. Journal of Graph Theory, 95(4), 655-676. https://doi.org/10.1002/jgt.22604
Balogh, J., Kostochka, A., Lavrov, M., & Liu, X. (2020). Long monochromatic paths and cycles in 2-edge-colored multipartite graphs. Moscow Journal of Combinatorics and Number Theory, 9(1), 55-100. https://doi.org/10.2140/moscow.2020.9.55