The erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

Preparation and characterization of molybdenum diselenide thin films

MoSe₂ layers, synthesized by annealing a molybdenum foil under selenium pressure, have been investigated by scanning electron microscopy, X-ray analysis, electron spectroscopy (XPS) and electrical resistance measurements. It has been found that stoichiometric layers are obtained after appropriate processing. The films crystallize in the hexagonal structure. The crystallites develop preferentially along the c axis. The binding energies deduced from the XPS lines were found to be in good agreement with those of the reference powder. The electrical resistance is governed by hopping conduction in the low temperature range (80–250 K) and by grain boundary scattering mechanisms at higher temperatures.     

Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif

A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.     

Maximal chains and cutsets of an ordered set: A menger type approach

We prove the following results which are related to Menger's theorem for (infinite) ordered sets. (i) If the space of maximal chains of an ordered set is compact, then the maximum number of pairwise disjoint maximal chains is finite and is equal to the minimum size of a cutset, (i.e. a set which meets all maximal chains). (ii) If the maximal chains pairwise intersect, then the intersection of finitely many is never empty. One corollary of (ii) is that, if the maximal chains pairwise intersect and if one of the maximal chains is complete, then there is an element common to all maximal chains.