LAUSR

It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering ≺ on their vertex set, and the function $${f_ \prec }\left( H \right) = \sup \left\{ {\chi \left( G \right)|G \in For{b_ \prec }\left( H \right)} \right\},$$f≺(H)=sup{χ(G)|G∈Forb≺(H)}, where Forb≺(H) denotes the set of all ordered graphs that do not contain a copy of H.If H contains a cycle, then as in the case of unordered graphs, f≺(H)=∞. However, in contrast to the unordered graphs, we describe an infinite family of ordered forests H with f≺(H) =∞. An ordered graph is crossing if there are two edges uv and u′v′ with u ≺ u′ ≺ v ≺ v′. For connected crossing ordered graphs H we reduce the problem of determining whether f≺(H) ≠∞ to a family of so-called monotonically alternating trees. For non-crossing H we prove that f≺(H) ≠∞ if and only if H is acyclic and does not contain a copy of any of the five special ordered forests on four or five vertices, which we call bonnets. For such forests H, we show that f≺(H)⩽2|V(H)| and that f≺(H)⩽2|V (H)|−3 if H is connected.