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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. [ABSTRACT FROM AUTHOR]

Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős–Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a P n -free subgraph of G (N , p) , practically for all values of N , n and p. Our work is also motivated by the recent progress on the size-Ramsey number of paths. [ABSTRACT FROM AUTHOR]