The famous Szemeredi–Trotter theorem states that any arrangement of n points and n lines in the plane determines $$O(n^{4/3})$$ incidences, and this bound is tight. In this paper, we prove the following Turan-type result for point-line incidence. Let $$\mathcal {L}_a$$ and $$\mathcal {L}_b$$ be two sets of t lines in the plane and let $$P=\{\ell _a \cap \ell _b : \ell _a \in \mathcal {L}_a, \,\ell _b \in \mathcal {L}_b\}$$ be the set of intersection points between $$\mathcal {L}_a$$ and $$\mathcal {L}_b$$ . We say that $$(P, \mathcal {L}_a \cup \mathcal {L}_b)$$ forms a natural $$t\times t$$ grid if $$|P| =t^2$$ , and $${\text {conv}}P$$ does not contain the intersection point of some two lines in $$\mathcal {L}_a$$ and does not contain the intersection point of some two lines in $$\mathcal {L}_b$$ . For fixed $$t > 1$$ , we show that any arrangement of n points and n lines in the plane that does not contain a natural $$t\times t$$ grid determines $$O(n^{{4}/{3}- \varepsilon })$$ incidences, where $$\varepsilon = \varepsilon (t)>0$$ . We also provide a construction of n points and n lines in the plane that does not contain a natural $$2 \times 2$$ grid and determines at least $$\Omega ({n^{1+{1}/{14}}})$$ incidences.

中文翻译：

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关于平面中点线排列的网格

著名的 Szemeredi-Trotter 定理指出，平面中 n 个点和 n 条线的任何排列都决定了 $$O(n^{4/3})$$ 的发生率，并且这个界限是紧的。在本文中，我们证明了以下点线入射的图兰型结果。令 $$\mathcal {L}_a$$ 和 $$\mathcal {L}_b$$ 是平面中的两组 t 线，并令 $$P=\{\ell _a \cap \ell _b : \ell _a \in \mathcal {L}_a, \,\ell _b \in \mathcal {L}_b\}$$ 是 $$\mathcal {L}_a$$ 和 $$\mathcal { 之间的交点集L}_b$$ 。我们说 $$(P, \mathcal {L}_a \cup \mathcal {L}_b)$$ 形成一个自然的 $$t\times t$$ 网格，如果 $$|P| =t^2$$ ，而 $${\text {conv}}P$$ 不包含 $$\mathcal {L}_a$$ 中某些两条线的交点，也不包含某些两条线的交点$$\mathcal {L}_b$$ 中的两行。对于固定 $$t > 1$$ ，我们表明，平面中不包含自然 $$t\times t$$ 网格的 n 个点和 n 条线的任何排列确定 $$O(n^{{4}/{3}- \varepsilon })$ $ 发生率，其中 $$\varepsilon = \varepsilon (t)>0$$ 。我们还提供了平面中 n 个点和 n 条线的构造，该构造不包含自然的 $$2 \times 2$$ 网格并确定至少 $$\Omega ({n^{1+{1}/{14} }})$$ 发生率。