More than forty years ago, Erdős conjectured that for any , every k-uniform hypergraph on n vertices without t disjoint edges has at most max edges. Although this appears to be a basic instance of the hypergraph Turán problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all . This improves upon the best previously known range , which dates back to the 1970s.

A causal set is a countably infinite poset in which every element is above finitely many others; causal sets are exactly the posets that have a linear extension with the order-type of the natural numbers; we call such a linear extension a natural extension. We study probability measures on the set of natural extensions of a causal set, especially those measures having the property of order-invariance: if we condition on the set of the bottom k elements of the natural extension, each feasible ordering among these k elements is equally likely. We give sufficient conditions for the existence and uniqueness of an order-invariant measure on the set of natural extensions of a causal set.

We discuss the connection between the expansion of small sets in graphs, and the Schatten norms of their adjacency matrices. In conjunction with a variant of the Azuma inequality for uniformly smooth normed spaces, we deduce improved bounds on the small-set isoperimetry of Abelian Alon–Roichman random Cayley graphs.

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

Any amicable pair ϕ, ψ of Sturmian morphisms enables aconstruction of a ternary morphism η which preserves the set of infinitewords coding 3-interval exchange. We determine the number of amicable pairs with the sameincidence matrix in SL±(2,ℕ) and we study incidence matricesassociated with the corresponding ternary morphisms η.

In this paper, we consider a class of scheduling problems that are among the fundamentaloptimization problems in operations research. More specifically, we deal with a particularversion called job shop scheduling with unit length tasks. Using theresults of Hromkovič, Mömke, Steinhöfel, and Widmayer presented in their work JobShop Scheduling with Unit Length Tasks: Bounds and Algorithms, we analyze theproblem setting for 2 jobs with an unequal number of tasks. We contribute a deterministicalgorithm which achieves a vanishing delay in certain cases and a randomized algorithmwith a competitive ratio tending to 1. Furthermore, we investigate the problem with 3 jobsand we construct a randomized online algorithm which also has a competitive ratio tendingto 1.

Let F3,3 be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all n ≥ 6, the maximum number of edges in an F3,3-free 3-graph on n vertices is . This sharpens results of Zhou [9] and of the second author and Rödl [7].