Supersaturation for Hypergraph-Weighted Independent Sets
arXiv:2607.14022
The paper develops supersaturation theorems for independent sets in a hypergraph whose size is measured by the number of edges they induce in another hypergraph, and applies these results to generalized Turán problems and additive combinatorial questions.
Abstract
Many extremal problems can be viewed as finding large independent sets in an auxiliary hypergraph. We propose a generalization of this by looking for ``large'' independent sets in a hypergraph where ``large'' is measured by how many edges induces in another hypergraph on the same vertex set as . We prove general supersaturation results for such extremal problems motivated by the breakthrough work of Ferber, McKinley and Samotij on counting -free graphs. As applications, we prove new supersaturation bounds for generalized Turán problems, as well as supersaturation bounds for a new set of extremal problems inspired by work of Fox and Pohoata on finding subsets which maximize the number of solutions to a given system of equations while avoiding solutions to another system.
22 pages, comments welcome!