We prove a bound of $O( k (n+m)\log^{d-1})$ on the number of incidences between $n$ points and $m$ axis parallel boxes in $\mathbb{R}^d$, if no $k$ boxes contain $k$ common points. That is, the incidence graph between the points and the boxes does not contain $K_{k,k}$ as a subgraph. This new bound improves over previous work by a factor of $\log^d n$, for $d >2$. We also study other variants of the problem. For halfspaces, using shallow cuttings, we get a near linear bound in two and three dimensions. Finally, we present near linear bound for the case of shapes in the plane with low union complexity (e.g. fat triangles).
翻译：我们证明美元(k(n+m)\log ⁇ d-1})是美元(n+m)\log ⁇ d-1}在美元点与美元($mathbb{R ⁇ d$)之间,如果没有美元盒子包含美元共同点,则美元($k$)与美元轴平行盒的数量为美元($mathbb{R ⁇ d$)之间。也就是说,点与框之间的发生率图中并不包含美元(käk),k}美元作为子图。这个新界限比以前的工作增加了一个系数($\log ⁇ d n$,$d>2$)。我们还要研究问题的其他变体。对于半空区,我们使用浅切,在两个和三个维度上接近线性线性。最后,我们提出近线性线性捆绑在平面上,其组合复杂性低(如脂肪三角)的形状(如脂肪三角形)的情况。
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