A recent paper by Balogh, Li, and Treglown initiated a study of the Dirac-type problem for ordered graphs. This paper will prove many results in this area. In particular, it asymptotically determines the minimum degree threshold for enforcement.

    (i) Full $H$ tiling of the ordered graph. For any fixed ordered graph $H$ with interval chromaticity at least $3$.

    (ii) the $H$ tiling of the ordered graph $G$ covering a constant fraction of the vertices of $G$ (for any fixed ordered graph $H$).

    (iii) $H$ cover of the ordered graph (for any fixed ordered graph $H$).

    The first two of these results solve the Balogh, Li, and Treglown problem while (iii) solving the Falgas-Ravry problem. (i) Note that in combination with the results of Balogh, Li, and Treglown, the asymptotic minimum degree threshold for enforcing perfect $H$ tiling is completely determined. Furthermore, in combination with the Balog, Li, and Tregroun theorems, we asymptotically set a minimum degree threshold for enforcing nearly perfect $H$ tiling on an ordered graph (any fixed ordered graph $H$). Prove the result that determines to Our work thus provides an ordered graph analogue of the influential tiling theorem of K’uhn and Osthus. [Combinatorica 2009] and of Comros [Combinatorica 2000]Each result shows some interesting and possibly unexpected behavior. Our solution to (i) makes use of novel absorption arguments.

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