Sobolev Spaces with only Trivial Isometries

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.     

A proof of the bounded graph conjecture

Summary An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. We prove an old conjecture of Halin, which characterizes the bounded graphs in terms of four forbidden topological subgraphs.Oblatum 17-IV-1991 & 25-X-1991     

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)². We conjecture that linear growth of f suffices to imply hamiltonicity.     

Dominating Functions and Graphs

A graph is called dominating if its vertices can be labelledwith integers in such a way that for every function : thegraph contains a ray whose sequence of labels eventually exceeds. We obtain a characterization of these graphs by producinga small family of dominating graphs with the property that everydominating graph must contain some member of the family.     

Abstract Separation Systems

Abstract separation systems provide a simple general framework in which both tree-shape and high cohesion of many combinatorial structures can be expressed, and their duality proved. Applications range from tangle-type duality and tree structure theorems in graphs, matroids or CW-complexes to, potentially, image segmentation and cluster analysis. This paper is intended as a concise common reference for the basic definitions and facts about abstract separation systems in these and any future papers using this framework.     

On the Cofinality of Infinite Partially Ordered Sets: Factoring a Poset into Lean Essential Subsets

We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable cofinality admitting such a decomposition are characterized by a forbidden substructure; the corresponding problem for uncountable cofinality remains open.     

An accessibility theorem for infinite graph minors

We prove the following recent conjecture of Halin. Let Γ₀ be the class of all graphs, and for every ordinal μ > 0 let Γ_μ be the class of all graphs containing infinitely many disjoint connected graphs from Γ_λ, for every λ < μ. Then a graph lies in all these classes Γ_μ if and only if it contains a subdivision of the infinite binary tree. Published by John Wiley & Sons, Inc., 2000 J Graph Theory 35: 273–277, 2000     

A note on weakly compact subsets in the projective tensor product of Banach spaces

Let X and Y be two Banach spaces. In this short note we show that every weakly compact subset in the projective tensor product of X and Y can be written as the intersection of finite unions of sets of the form , where K_X and K_Y are weakly compacts subsets of X and Y, respectively. If either X or Y has the Dunford–Pettis property, then any intersection of sets that are finite unions of sets of the form , where K_X and K_Y are weakly compact sets in X and Y, respectively, is weakly compact.