  It is known that every graph of sufficiently large chromatic number and
bounded clique number contains, as an induced subgraph, a subdivision of any
fixed forest, and a subdivision of any fixed cycle. Equivalently, forests and
triangles are pervasive, where H is pervasive (in some class of graphs) if for
all s>0, every graph in the class with bounded clique number and sufficiently
large chromatic number contains an induced subdivision of H, with every edge
subdivided at least s times.
  Which other graphs are pervasive? Chalopin, Esperet, Li and Ossona de Mendez
proved that every such graph is a forest of lanterns: roughly, the blocks are
lanterns (graphs obtained from a tree by adding one extra vertex), and there
are rules about how blocks fit together. It is not known whether every forest
of lanterns is pervasive; but in another paper two of us prove that banana
trees (multigraphs obtained from a forest by adding parallel edges) are
pervasive, thus generalizing the two results above. This paper contains the
first half of the proof, which works for any forest of lanterns, not just for
banana trees.
  A class of graphs is r-controlled if for every graph in the class, its
chromatic number is at most some function (determined by the class) of the
largest chromatic number of an r-ball in the graph. In this paper we prove that
for all r>1, every forest of lanterns is pervasive in every r-controlled class
  These results turn out particularly nicely when applied to string graphs
(intersection graphs of sets of curves in the plane). A chandelier is a graph
obtained from a tree by adding a vertex adjacent to its leaves. We prove that
the class of string graphs is 2-controlled, and thus forests of lanterns are
pervasive in this class. Furthermore, string graphs of sufficiently large
chromatic number and bounded clique number contain any fixed chandelier as an
induced subgraph.
