Tilings in vertex ordered graphs

Over recent years there has been much interest in both Turán and Ramsey properties of vertex ordered graphs. In this paper we initiate the study of embedding spanning structures into vertex ordered graphs. In particular, we introduce a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In the (unordered) graph setting, this problem was resolved by Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009]. We use our general framework to resolve the perfect $H$-tiling problem for all ordered graphs $H$ of interval chromatic number $2$. Already in this restricted setting the class of extremal examples is richer than in the unordered graph problem. In the process of proving our results, novel approaches to both the regularity and absorbing methods are developed.

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