AI 中文总结
该研究引入k-阈值概念，将Talagrand离散凸性猜想重新表述为k-阈值与期望阈值的比较关系，通过归约定理确定固定图k-阈值，证明k=2适用于部分生成图性质，还建立了低简并度目标图类的对应比较关系。
AI 中文摘要
我们引入了“k-阈值”的概念，证明Talagrand离散凸性猜想等价于：对于某个满足k≥2的通用整数，每个递增族的k-阈值至多为其期望阈值的通用常数倍。我们证明了一个归约定理，该定理通过图的合适分解中普通阈值来界定任意递增图性质的k-阈值。由此，我们基于自然k-密度参数，确定了每个固定图的k-阈值（误差在常数因子范围内）。我们还证明，对于若干经典生成图包含性质，k=2已足够；更一般地，对于目标图具有低简并度的广泛类别的图包含性质，我们建立了k-阈值与期望阈值之间的猜想比较关系。
英文摘要
We introduce the notion of "$k$-thresholds'' and show that Talagrand's discrete convexity conjecture is equivalent to the assertion that, for some universal integer $k \ge 2$, the $k$-threshold of every increasing family is at most a universal constant times its expectation threshold. We prove a reduction theorem that bounds the $k$-threshold of any increasing graph property in terms of ordinary thresholds of graphs in suitable decompositions of its members. As a consequence, we determine, up to a constant factor, the $k$-threshold of every fixed graph in terms of a natural $k$-density parameter. We also prove that $k=2$ suffices for several classical spanning graph containment properties. More generally, we establish the conjectured comparison between $k$-thresholds and expectation thresholds for broad classes of graph containment properties whose target graphs have low degeneracy.
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