Sub-Ramsey numbers for arithmetic progressions

Letm 3 andk 1 be two given integers. Asub-k-coloring of [n] = {1, 2,...,n} is an assignment of colors to the numbers of [n] in which each color is used at mostk times. Call an arainbow set if no two of its elements have the same color. Thesub-k-Ramsey number sr(m, k) is defined as the minimumn such that every sub-k-coloring of [n] contains a rainbow arithmetic progression ofm terms. We prove that((k – 1)m ²/logmk) sr(m, k) O((k – 1)m ² logmk) asm , and apply the same method to improve a previously known upper bound for a problem concerning mappings from [n] to [n] without fixed points.Research supported in part by Allon Fellowship and by a Bat Sheva de-Rothschild grant.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant No. 1-3-86-264.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

Polyethylenimine-Containing Cotton Fabrics

Abstract

Polyethylenimine-containing cotton fabrics were prepared by reaction of 1-epoxyethyl-3, 4-epoxycyclohexane with cotton fabric impregnated with polyethylenimine of a moderately high molecular weight. Fabrics with a high unremovable polyethylenimine content and with a different degree of cross-linking were obtained. Adsorption of several heavy metal salts such as mercuric chloride and cupric sulfate by the fabrics was investigated. Adsorption was controlled by polyethylenimine content of the fabric, extent of cross-linking, and the pH of the solution.     

Reaction of Lithium Cellulosate with Episulfides

The reactions of propylene, ethylene, and styrene sulfide with cotton fabrics in the form of their lithium cellulosate derivative, obtained by metallation with lithium methoxide, were studied. Both graft and homo-polymerization took place and in the reaction with propylene and ethylene sulfide, cotton fabrics with high polysulfide content were obtained. With propylene sulfide 50% of the monomer polymerized was grafted. The effects of monomer concentration, cellulosate degree of substitution, reaction time, and solvent on the polymerization were investigated. In contrast with propylene and ethylene sulfide, only small conversions were obtained in the reaction with styrene sulfide. The results were attributed to steric effects.     

Adsorption of Silver Nitrate by Cotton Fabrics Containing Polyethylene Sulfide

The adsorption of silver nitrate from aqueous solution by polyethylene sulfide-containing cotton fabrics was investigated. The adsorption could be described by a Freundlich-type isotherm with n= 1. δH^os was calculated from adsorption isotherms and from the equilibrium constant of complex formation and similar results of-5.6 and -5.4 kcal/mole were obtained, respectively. At low silver nitrate concentrations most of the silver nitrate present in solution could be adsorbed. It was found that adsorption by polyethylene sulfide in the fabric was much higher than that of homo-polyethylene sulfide.