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 the variant of the problem for points and halfspaces, where we use shallow cuttings to get a near linear bound in two and three dimensions.
翻译：我们证明美元(k(n+m)\log ⁇ d-1})是美元(n+m)和美元(m$)的结合值,以美元计点到美元($mathbb{R ⁇ d$)的轴平行盒的发生次数为美元($mathbb{R ⁇ d$),如果没有美元框包含美元(k$)的共同点。也就是说,点和框之间的事件图没有以美元作为子图。这个新的结合值比以前的工作改进了1美元($\log ⁇ d n$),以美元计值 > 2美元。我们还要研究点和半空格的问题变式,我们用浅的切口在两个维和三个维度上接近线性。
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