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An ordered graph H is a simple graph with a linear order on its vertex set. The corresponding Turán problem, first studied by Pach and Tardos, asks for the maximum number ex < (n, H) of edges in an ordered graph on n vertices that does not contain H as an ordered subgraph. It is known that ex < (n, H) > n 1+ε for some positive ε = ε(H) unless H is a forest that has a proper 2-coloring with one color class totally preceding the other one. Making progress towards a conjecture of Pach and Tardos, we prove that ex < (n, H) = n 1+o (1) holds for all such forests that are "degenerate" in a certain sense. This class includes every forest for which an n 1+o(1) upper bound was previously known, as well as new examples. Our proof is based on a density-increment argument.

a b s t r a c tAn n-by-n bipartite graph is H-saturated if the addition of any missing edge between its two parts creates a new copy of H. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a K s,s -saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a K (s,t) -saturated bipartite graph is n 2

A classic result of Erdős, Gyárfás and Pyber states that for every coloring of the edges of Kn with r colors, there is a cover of its vertex set by at most f(r)=O(r2logr) vertex‐disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of Kn but only on the number of colors. We initiate the study of this phenomenon in the case where Kn is replaced by the random graph G(n,p). Given a fixed integer r and p=p(n)≥n−1/r+ε, we show that with high probability the random graph G∼G(n,p) has the property that for every r‐coloring of the edges of G, there is a collection of f′(r)=O(r8logr) monochromatic cycles covering all the vertices of G. Our bound on p is close to optimal in the following sense: if p≪(logn/n)1/r, then with high probability there are colorings of G∼G(n,p) such that the number of monochromatic cycles needed to cover all vertices of G grows with n.

A graph H is K s -saturated if it is a maximal K s -free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K s -saturated graph was determined over 50 years ago by Zykov and independently by Erdős, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal K s -free subgraph of the Erdős-Rényi random graph G(n, p). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters.