We investigate prophet inequalities with competitive ratios approaching $1$, seeking to generalize $k$-uniform matroids. We first show that large girth does not suffice: for all $k$, there exists a matroid of girth $\geq k$ and a prophet inequality instance on that matroid whose optimal competitive ratio is $\frac{1}{2}$. Next, we show $k$-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio $1-O(\sqrt{\frac{\log k}{k}})$ for any $k$-fold matroid union. Our prophet inequality follows from an online contention resolution scheme. The key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone $1$-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a $1$-Lipschitz function that is not (approximately) self-bounding.

翻译：我们研究竞争比趋近于$1$的预言不等式，旨在推广$k$均匀拟阵。首先证明大围长条件并不充分：对任意$k$，总存在围长$\geq k$的拟阵及基于该拟阵的预言不等式实例，其最优竞争比为$\frac{1}{2}$。其次证明$k$重拟阵并具有充分性：针对任意$k$重拟阵并，我们给出了竞争比为$1-O(\sqrt{\frac{\log k}{k}})$的预言不等式。该预言不等式源于在线竞争消解方案，其关键技术要素是针对独立项上任意单调$1$-利普希茨函数提出的新型双准则集中不等式，该结论本身可能具有独立研究价值。将双准则集中不等式应用于我们的特定场景，可为非（近似）自界的$1$-利普希茨函数提供"切尔诺夫强度"的集中性保证。