A graph class is $\chi$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erd\H{o}s conjectured that intersection graphs of straight-line segments in the plane are $\chi$-bounded, but this was disproved by Pawlik et al. (2014), who showed another way to force large chromatic number in this class -- by triangle-free graphs $B_k$ with $\chi(B_k)=k$ constructed by Burling (1965). This also disproved the celebrated conjecture of Scott (1997) that classes of graphs excluding induced subdivisions of a fixed graph are $\chi$-bounded. We prove that in broad classes of graphs excluding induced subdivisions of a fixed graph, including the increasingly more general classes of segment intersection graphs, string graphs, region intersection graphs, and hereditary classes of graphs with finite asymptotic dimension, large chromatic number can be forced only by large cliques or large graphs $B_k$. One corollary is that the hereditary closure of $\{B_k\colon k\geq 1\}$ forms a minimal hereditary graph class with unbounded chromatic number -- the second known graph class with this property after the class of complete graphs. Another corollary is that the decision variant of approximate coloring in the aforementioned graph classes can be solved in polynomial time by exhaustively searching for a sufficiently large clique or copy of $B_k$. We also discuss how our results along with some results of Chudnovsky, Scott, and Seymour on the existence of colorings can be turned into polynomial-time algorithms for the search variant of approximate coloring in string graphs (with intersection model in the input) and other aforementioned graph classes. Such an algorithm has not yet been known for any graph class that is not $\chi$-bounded.

翻译：一个图类被称为$\chi$-有界的，如果在该类图中迫使色数增大的唯一方式是形成大团。在20世纪70年代，Erdős猜想平面中直线段的交图是$\chi$-有界的，但这一猜想被Pawlik等人（2014）证伪，他们展示了在该类图中迫使大色数的另一种方式——通过使用Burling（1965）构造的、满足$\chi(B_k)=k$的无三角形图$B_k$。这也证伪了Scott（1997）的著名猜想，即排除固定图的诱导细分的图类是$\chi$-有界的。我们证明，在排除固定图诱导细分的广泛图类中——包括日益广义的线段交图类、弦图类、区域交图类以及具有有限渐近维度的遗传图类——大色数只能由大团或大图$B_k$迫使。一个推论是$\{B_k\colon k\geq 1\}$的遗传闭包构成了具有无界色数的最小遗传图类——这是继完全图类之后第二个已知具有此性质的图类。另一个推论是，在上述图类中，近似着色的判定变体可以通过穷举搜索足够大的团或$B_k$的副本在多项式时间内求解。我们还讨论了如何将我们的结果与Chudnovsky、Scott和Seymour关于着色存在性的一些结果相结合，转化为弦图（输入中给出交模型）及其他上述图类中近似着色的搜索变体的多项式时间算法。对于任何非$\chi$-有界的图类，此类算法此前尚未已知。