An $r$-quasiplanar graph is a graph drawn in the plane with no $r$ pairwise crossing edges. We prove that there is a constant $C>0$ such that for any $s>2$, every $2^s$-quasiplanar graph with $n$ vertices has at most $n(\frac{C\log n}{s})^{2s-4}$ edges. A graph whose vertices are continuous curves in the plane, two being connected by an edge if and only if they intersect, is called a string graph. We show that for every $\epsilon>0$, there exists $\delta>0$ such that every string graph with $n$ vertices, whose chromatic number is at least $n^{\epsilon}$ contains a clique of size at least $n^{\delta}$. A clique of this size or a coloring using fewer than $n^{\epsilon}$ colors can be found by a polynomial time algorithm in terms of the size of the geometric representation of the set of strings. In the process, we use, generalize, and strengthen previous results of Lee, Tomon, and others. All of our theorems are related to geometric variants of the following classical graph-theoretic problem of Erd\H os, Gallai, and Rogers. Given a $K_r$-free graph on $n$ vertices and an integer $s<r$, at least how many vertices can we find such that the subgraph induced by them is $K_s$-free?

翻译：美元平面图是一张在平面上绘制的图表,没有美元平面交叉边缘。我们证明有一个固定的 $C>0 美元,对于任何$>2美元,每2美元,每2美元平面图,每2美元平面图,有1美元正方平面,有1美元正方平面,有1美元正方平面。一个图,其顶端是平面上连续的曲线,两张正平面曲线连接,两张平面图,如果是一个平面,只有它们相互交错,就被称为字符平面图。我们显示,每1美元正平面图,每张每张正平面图都有1美元,每张正平面图至少有1美元正方平面大小。这种大小或彩色的曲线,在直面上方平面的平面上可以找到一个最不固定的平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。