Induced subgraphs of graphs with large chromatic number. V. Chandeliers and strings

It is known that every graph of sufficiently large chromatic number and bounded clique contains, as an induced subgraph, a subdivision any fixed forest, cycle. Equivalently, forest pervasive, K3 in the class all graphs, where we say H “pervasive” (in some graphs) if for ℓ≥1, has subgraph H, which edge replaced by path at least ℓ edges. Which other graphs are pervasive? was proved Chalopin, Esperet, Li Ossona de Mendez such “forest lanterns”: roughly, block “lantern”, obtained from tree adding one extra vertex, there rules about how blocks fit together. not whether lanterns pervasive graphs; but another paper two us prove “banana trees” is, multigraphs parallel edges, thus generalizing results above. This contains first half proof, works lanterns, just banana trees. Say “ρ-controlled” class, its most function (determined class) largest ρ-ball graph. In this ρ≥2, ρ-controlled class. These turn out particularly nicely when applied to string graphs. A “chandelier” special lantern, vertex adjacent precisely leaves tree. “string graph” intersection set curves plane. There with arbitrarily large. We 2-controlled, consequently class; fact something stronger true, each chandelier (not subdivision); same forests chandeliers (there condition on attached together).