Duality and $\chi^<$-Boundedness of Ordered Graphs

We show that there exists only one duality pair for ordered graphs. We will also define a corresponding definition of $\chi^<$-boundedness for ordered graphs and show that all ordered graphs are $\chi^<$-bounded and prove an analogy of Gy\'arf\'as-Sumner conjecture for ordered graphs. We also prove an analogy of Sparse Incomparability Lemma for ordered graphs. We then use this result to show classes of ordered graphs that form a dense order under ordered homomorphisms. We also show that compared to graphs, ordered graphs have more gaps, defined by consecutive monotone matchings and by even more generic pairs of ordered graphs differing by one isolated edge.