Bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter $k\ge 2$, items having sizes in $(0,1]$ associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem and prove a lower bound of 2 on the overall asymptotic competitive ratio. Additionally, we significantly improve the known lower bounds on the asymptotic competitive ratio for every specific value of k. The novelty of our constructions is based on full adaptivity that creates large gaps between item sizes. Last, we show a lower bound strictly larger than 2 on the asymptotic competitive ratio of the online 2-dimensional vector packing problem, where no such lower bound was known even for fixed high dimensions.

具有基数约束的装箱是基本装箱问题。在带有参数的在线版本中$ķ\ge 2$，尺寸为 $（0，\mathrm{1个}]$与它们相关联的组件被逐一呈现，并被打包到单位容量的存储箱中，这样就不会超过存储箱的容量，并且没有一个存储箱接收超过k个项目。我们解决了在线问题，并证明了整体渐近竞争率的下限2。此外，对于k的每个特定值，我们显着改善了渐近竞争比的已知下界。我们结构的新颖性基于完全的适应性，从而在物品尺寸之间造成了巨大的差距。最后，我们在在线二维向量打包问题的渐近竞争比上显示了一个严格大于2的下界，即使对于固定的高尺寸，也没有这样的下界。