The dimension of the finite subsets κ

We answer a question of M. Pouzet by showing that the Dushnik-Miller dimension of the finite subsets of the infinite cardinal ordered by inclusion is ().This paper was written when the first author was visiting the University of Calgary, June, 1995. Research partially supported by Office of Naval Research grant NOOO14-90-1206.Research supported by NSERC grant #69-0982.     

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.     

Dithiadiazolium salts as initiators for the cationic ring-opening polymerization of tetrahydrofuran

Mono-, bis- and tris-(1,3,2,4-dithiadiazolium) salts [R-(CNSNS ⁺)_n]_n+[AsF^-₆]_n (R = aryl, n = 1, 2, 3) were found to initiate the cationic ring-opening polymerization of tetrahydrofuran (THF) at room temperature to give clear gels from which the pure polymer was precipitated. 1,3,2,4-Dithiadiazolium cations associated with the hard [AsF₆]^- anion thus constitute a new class of cationic polymerization initiators. The poly(THF) formed by initiation with 1,3,2,4-dithiadiazolium cation was characterized by gel permeation chromatography, infrared spectrophotometry, and ¹³C-NMR spectroscopy. Number-average molecular weights of 198 700 g mol^-1 (polydispersity 1.96) and 190 000 g mol^-1 (polydispersity 1.61) were obtained using [PhCNSNS ] [AsF₆] and [C₆H₃-1,3,5-(CNSNS )₃][AsF₆]₃, respectively, as initiators. The use of multifunctional dithiadiazolium salts as initiators suggests that they may be useful in the preparation of starburst and dendritic polymers. © 1992 John Wiley & Sons, Inc.