Transformations of the transfinite plane

We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every inaccessible cardinal $\kappa $ , if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $ , there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$ , there exist $x\neq y$ in X such that $f(x+y)=g$ .

THE POLARISED PARTITION RELATION FOR ORDER TYPES

We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.

A TAIL CONE VERSION OF THE HALPERN–LÄUCHLI THEOREM AT A LARGE CARDINAL

The classical Halpern–Läuchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height ω, there exist strong subtrees sharing the same level set such that tuples in the product of the strong subtrees consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern–Läuchli theorem at a large cardinal (see Theorem 3.1), which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.

Random reals and polarized colorings

We analyze the strong polarized partition relation with respect to several cardinal characteristics and forcing notions of the reals. We prove that random reals (as well as the existence of real-valued measurable cardinals) yield downward negative polarized relations.

THE TOPOLOGICAL PIGEONHOLE PRINCIPLE FOR ORDINALS

Given a cardinal κ and a sequence ${\left( {{\alpha _i}} \right)_{i \in \kappa }}$ of ordinals, we determine the least ordinal β (when one exists) such that the topological partition relation$$\beta \to \left( {top\,{\alpha _i}} \right)_{i \in \kappa }^1$$holds, including an independence result for one class of cases. Here the prefix “top” means that the homogeneous set must be of the correct homeomorphism class rather than the correct order type. The answer is linked to the nontopological pigeonhole principle of Milner and Rado.

Study on Dyeing Properties of Chrysophanol on Poly(trimethylene terephthalate)Fiber

In order to provide the theoretical basis for synthetic fiber dyeing with natural dyes, the dyeing properties of chrysophanol on Poly(trimethylene terephthalate)(PTT) fiber were investigated. The appropriate dyeing condition was 0.5g•dm-3 Paregal O in the dyebath, pH=5 and the final temperature at 100°C~110°C. The dye uptake of chrysophanol on PTT fiber was relatively small, and the building-up property of chrysophanol on PTT fiber was poor. The adsorption of chrysophanol on PTT fiber accorded with Nerst Partition Relation. The partition coefficient decreased with the temperature rising while the dyeing affinity (-△μ0) increased with the temperature rising.

Dyeing Behavior of Chrysophanol on Polyester Fiber

The dyeing properties of chrysophanol on polyester (PET) were investigated. Chrysopahnol was stable with the pH range of 3 to 7. And in this pH range, the hue of Chrysophanol of dyed polyester showed very little change. The K/S values increased significantly when the temperature was raised from 90°C to 110°C, and if the temperature was 110°C, the dyeing time exceeded 90min, the dye uptake didn’t increase any more. The adsorption of chrysophanol on polyester accorded with Nerst Partition Relation, and the adsorption capacity increased linearly with the increasing chrysophanol concentration. The partition coefficient and affinity increased with the temperature rising.

Weak square bracket relations for Pκ(λ)

We study the partition relation that is a weakening of the usual partition relation . Our main result asserts that if κ is an uncountable strongly compact cardinal and , then does not hold.