Jozsef Balogh

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.