Two point set extensions- a counterexample

Hausdorff Measures and Two Point Set Extensions

We investigate the question: under which conditions is a-compact partial two point contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a suucient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any-compact partial two point set such that ...

Hamiltonicity in Locally Finite Graphs: Two Extensions and a Counterexample

We state a sufficient condition for the square of a locally finite graph to contain a Hamilton circle, extending a result of Harary and Schwenk about finite graphs. We also give an alternative proof of an extension to locally finite graphs of the result of Chartrand and Harary that a finite graph not containing K4 or K2,3 as a minor is Hamiltonian if and only if it is 2-connected. We show furth...

Measures and Two Point Set

We investigate the question: under which conditions is a-compact partial two point contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a suucient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any-compact partial two point set such that ...

A counterexample to the dominating set conjecture

The metric polytope metn is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities xij − xik − xjk ≤ 0 and xij + xik + xjk ≤ 2 for all triples i, j, k of {1, . . . , n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, i...

Weak Compactness of the Set of ε-Extensions