In discrepancy theory, the basic question is whether a structure can be partitioned in a balanced way, or if there is always some “discrepancy” no matter how the partition is made. In the context of graph theory, a well-studied question is whether for a given host graph, any 2-coloring of its edges must contain a certain substructure "with high discrepancy", meaning that within this substructure one of the color classes is significantly larger than the other. Classical results include the works of Erdos and Spencer on cliques and of Erdos, Füredi, Loebl and Sos on spanning trees. In this talk, I will present some recent results for spanning substructures of hypergraphs such as perfect matchings, tight Hamilton cycles and Steiner triple systems.

Based on joint works with Lior Gishboliner, Peleg Michaeli and Amedeo Sgueglia.