Given a family of squares in the plane, their $packing \ problem$ asks for the maximum number, $\nu$, of pairwise disjoint squares among them, while their $hitting \ problem$ asks for the minimum number, $\tau$, of points hitting all of them, $\tau \ge \nu$. Both problems are NP-hard even if all the rectangles are unit squares and their sides are parallel to the axes. The main results of this work are providing the first bounds for the $\tau / \nu$ ratio on not necessarily axis-parallel squares. We establish an upper bound of $6$ for unit squares and $10$ for squares of varying sizes. The worst ratios we can provide with examples are $3$ and $4$, respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval $[\frac{3}{2},2]$ for unit squares and $[\frac{3}{2},4]$ for arbitrary squares. The new bounds necessitate a mixture of novel and classical techniques of possibly extendable use. Furthermore, we study rectangles with a bounded ``aspect ratio'', where the $aspect \ ratio$ of a rectangle is the larger side of a rectangle divided by its smaller side. We improve on the well-known best $\tau/\nu$ bound, which is quadratic in terms of the aspect ratio. We reduce it from quadratic to linear for rectangles, even if they are not axis-parallel, and from linear to logarithmic, for axis-parallel rectangles. Finally, we prove similar bounds for the chromatic numbers of squares and rectangles with a bounded aspect ratio.

翻译：鉴于平面上的方形, 他们的 $ 包装 和 问题 的 比例 要求 最大 数, $\ nu$ 的 比例, 而他们的 $ 和 问题 $ 的 问题 要求 最小数, $ tau\ 美元, 所有点的最小数, $ tau\ ge\ nu$ 。 这两个问题都是 NP 硬 的, 即使所有的矩形都是单位方形, 它们的两面都与轴相平行 。 这项工作的主要结果是 $\ tau / nu$ 的最大值, 不一定是轴- 线性平面的 美元 。 我们为单位方/ / 和 $ 线性平面上的 $ 6 美元 和 10 美元 的上限 。 最坏的比 3 和 4 。 在轴- 方圆形 的 比较中, 所考虑的 比例是, 我们的平面和 正在变方 的 的正方 的 直方 。