A r-uniform hypergraph H (or a r-graph, for short) is a collection E= E(H) of r-element subsets (called edges) of a set V = V(H) (called vertices) . We say a r-graph H is (n, e)-unavoidable if every r-graph with n vertices and e edges must contain H . In this paper we investigate the largest possible number of edges in an (n, e)-unavoidable 3-graph for fixed n and e. We also study the structure...

We prove that every separable metric space which admits an `1tree as a Lipschitz quotient, has a σ-porous subset which contains every Lipschitz curve up to a set of 1-dimensional Hausdorff measure zero. This applies to any Banach space containing `1. We also obtain an infinite-dimensional counterexample to the Fubini theorem for the σ-ideal of σ-porous sets.

A connected graph is double-connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), deened on the set of integers exceeding three, such that every double-connected graph on at least h(n) vertices must contain, as an induced subgraph, a special double-connected graph, which is either one of the following graphs or th...

It is common practice to partition complex workflows into separate channels in order to speed up their completion times. When this is done within a distributed environment, unavoidable fluctuations make individual realizations depart from the expected average gains. We present a method for breaking any complex workflow into several workloads in such a way that once their outputs are joined, the...

Balogh and Bollobás [Combinatorica 25, 2005] prove that for any k there is a constant f(k) such that any set system with at least f(k) sets reduces to a k-star, an k-costar or an k-chain. They proved f(k) < (2k)2 k . Here we improve it to f(k) < 2ck 2 for some constant c > 0. This is a special case of the following result on the multi-coloured forbidden configurations at 2 colours. Let r be giv...

We complete the classification of binary patterns in partial words that was started in earlier publications by proving that the partial word avoidability index of the binary pattern ABABA is two and the one of the binary pattern ABBA is three.