AI 中文总结
研究在超图中寻找加权独立集的问题，基于Ferber等人的工作证明一般超饱和结果，应用于广义Turán问题及受Fox和Pohoata工作启发的新极值问题，给出相关超饱和界。
AI 中文摘要
许多极值问题可视为在辅助超图中寻找大的独立集。我们对此进行推广，在超图\(\mathcal{F}\)中寻找‘大’独立集\(I\)，‘大’由\(I\)在与\(\mathcal{F}\)有相同顶点集的另一超图\(\mathcal{H}\)中诱导出的边数衡量。受Ferber、McKinley和Samotij关于不含\(F\)图计数的突破性工作启发，我们证明了此类极值问题的一般超饱和结果。作为应用，我们证明了广义Turán问题的新超饱和界，以及受Fox和Pohoata关于寻找最大化给定方程组解的数量同时避免另一方程组解的子集\(A\subseteq\mathbb{N}\)的工作启发的一组新极值问题的超饱和界。
英文摘要
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 $I$ in a hypergraph $\mathcal{F}$ where ``large'' is measured by how many edges $I$ induces in another hypergraph $\mathcal{H}$ on the same vertex set as $\mathcal{F}$. We prove general supersaturation results for such extremal problems motivated by the breakthrough work of Ferber, McKinley and Samotij on counting $F$-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 $A\sub\mathbb{N}$ which maximize the number of solutions to a given system of equations while avoiding solutions to another system.
Comments22 pages, comments welcome!
