The goal of this paper is to give a new, abstract approach to cover-decomposition and polychromatic colorings using hypergraphs on ordered vertex sets. We introduce an abstract version of a framework by Smorodinsky and Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called ABA-free hypergraphs, which are a generalization of interval graphs. Using our methods, we prove that (2k − 1)-uniform ABA-free hypergraphs have a polychromatic k-coloring, a problem posed by the second author. We also prove the same for hypergraphs defined on a point set by pseudohalfplanes. These results are best possible. We could only prove slightly weaker results for dual hypergraphs defined by pseudohalfplanes, and for hypergraphs defined by pseudohalfspheres. We also introduce another new notion that seems to be important for investigating polychromatic colorings and ǫ-nets, shallow hitting sets. We show that all the above hypergraphs have shallow hitting sets, if their hyperedges are containment-free.

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) linear if its extremal function is O (n). Our main results are modest steps towards the elusive goal of characterizing linear patterns. We find novel ways to generate new linear patterns from known ones and use this to prove the linearity of some patterns. We also find the first minimal non-linear pattern of weight above 4. We also propose an infinite sequence of patterns that we conjecture to be minimal non-linear but have Ω(n log n) as their extremal function. We prove a weaker statement only, namely that there are infinitely many minimal not quasi-linear patterns among the submatrices of these matrices. For the definition of these terms see below.

We settle a problem of Dujmović, Eppstein, Suderman, and Wood by showing that there exists a function f with the property that every planar graph G with maximum degree d admits a drawing with noncrossing straight-line edges, using at most f (d) different slopes. If we allow the edges to be represented by polygonal paths with one bend, then 2d slopes suffice. Allowing two bends per edge, every planar graph with maximum degree d ≥ 3 can be drawn using segments of at most ⌈d/2⌉ different slopes. There is only one exception: the graph formed by the edges of an octahedron is 4-regular, yet it requires 3 slopes. These bounds cannot be improved.

A family F ⊆ 2 [n] saturates the monotone decreasing property P if F satisfies P and one cannot add any set to F such that property P is still satisfied by the resulting family. We address the problem of finding the minimum size of a family saturating the k-Sperner property and the minimum size of a family that saturates the Sperner property and that consists only of l-sets and (l + 1)-sets.