The Burling sequence is a sequence of triangle-free graphs of unbounded chromatic number. The class of Burling graphs consists of all the induced subgraphs of the graphs of this sequence. In the first and second parts of this work, we introduced derived graphs, a class of graphs, equal to the class of Burling graphs, and proved several geometric and structural results about them. In this third part, we use those results to find some Burling and non-Burling graphs, and we see some applications of this in the theory of $\chi$-boundedness. In particular, we show that several graphs, like $ K_5 $, some series-parallel graphs that we call necklaces, and some other graphs are not weakly pervasive.

翻译：Burling序列是一类色数无界的无三角形图序列。Burling图类由该序列中所有图的诱导子图构成。在本工作的第一部分和第二部分中，我们引入了导出图（derived graphs）这一概念——该类图与Burling图类等价——并证明了若干几何与结构性质。在第三部分中，我们利用这些结果发现了某些Burling图和非Burling图，并探讨了它们在$\chi$-有界性理论中的若干应用。特别地，我们证明了某些图，如完全图$K_5$、被称为“项链图”（necklaces）的某些串并联图以及其他一些图，并非弱渗透图（weakly pervasive）。