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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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