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 \emph{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. For every $r\ge 3$, we construct families of $n$ segments in the plane without $r$ pairwise crossing members, which have the property that in any coloring of the segments with fewer than $c \log\log n $ colors, at least one of the color classes contains $r-1$ pairwise crossing segments. Here $c=c(r)>0$ is a suitable constant. In the process, we use, generalize, and strengthen previous results of Lee, Tomon, Walczak, 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,没有美元双向交叉边缘。我们证明有一个固定的 $C>0 美元,对于任何$>2美元,每个2美元平面图,加上美元正方平面图,最多为$(frac{C\log n ⁇ s}) $2-4}。一个平面的螺旋是连续曲线的图,两个在交错时只能用边缘连接?一个叫做\emph{string group$。我们证明,对于每个美元正方平面图,每个$2美元每张平面图,每个平面图,其色数至少为$n ⁇ delta}包含一个大小的螺旋。这种大小或颜色比美元平面平面的平面值更小,对于一个多色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。