A string graph is the intersection graph of curves in the plane. We prove that for every \(\epsilon >0\), if G is a string graph with n vertices such that the edge density of G is below \({1}/{4}-\epsilon \), then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than \({1}/{4}+\epsilon \) such that there is an edge connecting any two logarithmic sized subsets of the vertices. The existence of linear sized sets A and B with no edges between them in sufficiently sparse string graphs is a direct consequence of a recent result of Lee about separators. Our main theorem finds the largest possible density for which this still holds. In the special case when the curves are x-monotone, the same result was proved by Pach and the author of this paper, who also proposed the conjecture for the general case.

字符串图是平面中曲线的交集图。我们证明对于每个\(\epsilon >0\)，如果G是一个具有n个顶点的字符串图，使得 G的边密度低于\({1}/{4}-\epsilon \)，那么V ( G ) 包含两个线性大小的子集A和 B，它们之间没有边。常数 1/4 是这种现象的一个尖锐阈值，因为存在边密度小于\({1}/{4}+\epsilon \)的字符串图，因此存在连接任意两个对数大小的子集的边顶点。存在线性大小的集合A和 在足够稀疏的字符串图中，它们之间没有边的B是 Lee 最近关于分隔符的结果的直接结果。我们的主要定理找到了这仍然成立的最大可能密度。在曲线为x -单调的特殊情况下，Pach 和本文作者也证明了相同的结果，他也提出了一般情况下的猜想。