Partition Relations Research Articles

THE NEXT BEST THING TO A P-POINT

AbstractWe study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

Rootlshoot partitioning and water relations in <i>Qualea grandiflora </i>(Vochysiaceae) seedlings under water stress

Se cultivó plántulas de Qualea granditlóra a partir de semillas de "Cerrado", con distintas cantidades de agua, en invernadero o cámara de crecimiento, eligiendo 5 o 10 plantas al azar. Reaccionaron al limitado aporte hídrico disminuyendo la conductividad estomática y los potenciales hídrico y de turgencia. En las plantas bajo deficit hídrico no se observó ajuste osmótico. La relación raiz: vástago fue más alta bajo estrés hídrico, presentando la parte aérea menor crecimiento debido a la menor producción y expansión foliares.

From Edge-Coloring to Strong Edge-Coloring

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: $k$-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian. In this coloring, the set $S(v)$ of colors used by edges incident to a vertex $v$ does not intersect $S(u)$ on more than $k$ colors when $u$ and $v$ are adjacent. We provide some sharp upper and lower bounds for $\chi'_{k\text{-int}}$ for several classes of graphs. For $l$-degenerate graphs we prove that $\chi'_{k\text{-int}}(G)\leq (l+1)\Delta -l(k-1)-1$. We improve this bound for subcubic graphs by showing that $\chi'_{2\text{-int}}(G)\leq 6$. We show that calculating $\chi'_{k\text{-int}}(K_n)$ for arbitrary values of $k$ and $n$ is related to some problems in combinatorial set theory and we provide bounds that are tight for infinitely many values of $n$. Furthermore, for complete bipartite graphs we prove that $\chi'_{k\text{-int}}(K_{n,m}) = \left\lceil \frac{mn}{k}\right\rceil$. Finally, we show that computing $\chi'_{k\text{-int}}(G)$ is NP-complete for every $k\geq 1$.An addendum was added to this paper on Jul 4, 2015.

CH, [formula omitted], disintegration of measures, and [formula omitted] sets

In 1950 Maharam asked whether every disintegration of a σ-finite measure into σ-finite measures is necessarily uniformly σ-finite. Over the years under special conditions on the disintegration, the answer was shown to be yes. We show here that the question is equivalent to the existence of a Borel uniformization of a certain set defined from the disintegration. Moreover, we show that the answer may depend on the axioms of set theory in the following sense. If CH, the continuum hypothesis holds, then the answer is no. Our proof of this leads to some interesting problems in infinitary combinatorics. Also, if Gödel's axiom of constructibility V=L holds, then not only is the answer no, but of equal interest is the construction of Π11 sets with very special properties.

Set Theory

This stimulating workshop featured a broad selection of some of the most important recent developments in combinatorial set theory, the theory and applications of forcing, large cardinal theory and descriptive set theory.

Serial compact spaces and the maximal spectrum of a Bézout domain

Serial quasi-compact T1-spaces are exactly the spaces that arise as maximal spectra of Bézout domains. The closed unit interval is serial. A nested sequence of directed graphs is constructed to show that the unit square is still serial. Using arguments of combinatorial set theory, it is proved that big powers of the unit interval are not serial.

Idiosynchromatic Poetry

We prove equivalences of asymmetric partition relations involving natural numbers and products of weakly compact cardinals ?, infinite cardinals ?<? and natural numbers to certain classes of finitary problems in the theory of edge-coloured digraphs. We are able to determine three classes of Ramsey numbers exactly, typical examples in these classes are r(? 22,3), r(??2,3) and r(??3,3). We moreover provide general upper bounds for r(? 2 m,3) and r(??m,n).

A Theory of Stationary Trees and the Balanced Baumgartner–Hajnal–Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we describe a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Then \[ \text{non-$\left(2^{<\kappa}\right)$-special tree } \to \left(\kappa + \xi \right)^2_k. \] This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{<\kappa})^+$, the simplest example of a non-$(2^{<\kappa})$-special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem: Let $\kappa$ be any infinite regular cardinal, let $\xi$ be any ordinal such that $2^{\left|\xi\right|} < \kappa$, and let $k$ be any natural number. Let $P$ be a partially ordered set such that $P \to (2^{<\kappa})^1_{2^{<\kappa}}$. Then \[ P \to \left(\kappa + \xi \right)^2_k. \]

Some logically weak Ramseyan theorems

We study four families of consequences of Ramsey's Theorem from the viewpoint of reverse mathematics. The first, which we call the Achromatic Ramsey Theorem, is from a partition relation introduced by Erdős, Hajnal and Rado: ω→[ω]c,≤dr, which asserts that for every f:[ω]r→c there exists an infinite H with |f([H]r)|≤d. The second and third are the Free Set Theorem and the Thin Set Theorem, which were introduced by Harvey Friedman. And the last is the Rainbow Ramsey Theorem. We show that, most theorems from these families are quite weak, i.e., they are strictly weaker than ACA0 over RCA0. Interestingly, these families turn out to be closely related. We establish the so-called strong cone avoidance property of most instances of the Achromatic Ramsey Theorem by an induction of exponents, then apply this and a similar induction to obtain the strong cone avoidance property of the Free Set Theorem. From the strong cone avoidance property of the Achromatic Ramsey Theorem and the Free Set Theorem, we derive the strong cone avoidance property of the Thin Set Theorem and the Rainbow Ramsey Theorem. It follows easily that a theorem with the strong cone avoidance property does not imply ACA0 over RCA0.

Quantifying heterogeneity attributable to polythetic diagnostic criteria: theoretical framework and empirical application.

Heterogeneity within psychiatric disorders is both theoretically and practically problematic: For many disorders, it is possible for 2 individuals to share very few or even no symptoms in common yet share the same diagnosis. Polythetic diagnostic criteria have long been recognized to contribute to this heterogeneity, yet no unified theoretical understanding of the coherence of symptom criteria sets currently exists. A general framework for analyzing the logical and mathematical structure, coherence, and diversity of Diagnostic and Statistical Manual diagnostic categories (DSM-5 and DSM-IV-TR) is proposed, drawing from combinatorial mathematics, set theory, and information theory. Theoretical application of this framework to 18 diagnostic categories indicates that in most categories, 2 individuals with the same diagnosis may share no symptoms in common, and that any 2 theoretically possible symptom combinations will share on average less than half their symptoms. Application of this framework to 2 large empirical datasets indicates that patients who meet symptom criteria for major depressive disorder and posttraumatic stress disorder tend to share approximately three-fifths of symptoms in common. For both disorders in each of the datasets, pairs of individuals who shared no common symptoms were observed. Any 2 individuals with either diagnosis were unlikely to exhibit identical symptomatology. The theoretical and empirical results stemming from this approach have substantive implications for etiological research into, and measurement of, psychiatric disorders.

Combinatorial dichotomies and cardinal invariants

Assuming the P-ideal dichotomy, we attempt to isolate those cardinal char- acteristics of the continuum that are correlated with two well-known consequences of the proper forcing axiom. We find a cardinal invariant x such that the statement that x >ω 1 is equivalent to the statement that 1, ω, ω1, ω × ω1 ,a nd ( ω1) <ω are the only cofinal types of directed sets of size at most ℵ1. We investigate the corresponding prob- lem for the partition relation ω1 → (ω1 ,α ) 2 for all α<ω 1. To this effect, we investigate partition relations for pairs of comparable elements of a coherent Suslin tree S.W e show that a positive partition relation for such pairs follows from the maximal amount of the proper forcing axiom compatible with the existence of S. As a consequence, we conclude that after forcing with the coherent Suslin tree S over a ground model satisfying this relativization of the proper forcing axiom, ω1 → (ω1 ,α ) 2 for all α<ω 1 .W e prove that this positive partition relation for S cannot be improved by showing in ZFC that S �→ (ℵ1 ,ω +2 ) 2 .

A conjugate class of random probability measures based on tilting and with its posterior analysis

This article constructs a class of random probability measures based on exponentially and polynomially tilting operated on the laws of completely random measures. The class is proved to be conjugate in that it covers both prior and posterior random probability measures in the Bayesian sense. Moreover, the class includes some common and widely used random probability measures, the normalized completely random measures (James (Poisson process partition calculus with applications to exchangeable models and Bayesian nonparametrics (2002) Preprint), Regazzini, Lijoi and Pr\"{u}nster (Ann. Statist. 31 (2003) 560-585), Lijoi, Mena and Pr\"{u}nster (J. Amer. Statist. Assoc. 100 (2005) 1278-1291)) and the Poisson-Dirichlet process (Pitman and Yor (Ann. Probab. 25 (1997) 855-900), Ishwaran and James (J. Amer. Statist. Assoc. 96 (2001) 161-173), Pitman (In Science and Statistics: A Festschrift for Terry Speed (2003) 1-34 IMS)), in a single construction. We describe an augmented version of the Blackwell-MacQueen P\'{o}lya urn sampling scheme (Blackwell and MacQueen (Ann. Statist. 1 (1973) 353-355)) that simplifies implementation and provide a simulation study for approximating the probabilities of partition sizes.

Comparisons of Polychromatic and Monochromatic Ramsey Theory

AbstractWe compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on ω.

Quantitation of small intestinal permeability during normal human drug absorption.

BackgroundUnderstanding the quantitative relationship between a drug’s physical chemical properties and its rate of intestinal absorption (QSAR) is critical for selecting candidate drugs. Because of limited experimental human small intestinal permeability data, approximate surrogates such as the fraction absorbed or Caco-2 permeability are used, both of which have limitations.MethodsGiven the blood concentration following an oral and intravenous dose, the time course of intestinal absorption in humans was determined by deconvolution and related to the intestinal permeability by the use of a new 3 parameter model function (“Averaged Model” (AM)). The theoretical validity of this AM model was evaluated by comparing it to the standard diffusion-convection model (DC). This analysis was applied to 90 drugs using previously published data. Only drugs that were administered in oral solution form to fasting subjects were considered so that the rate of gastric emptying was approximately known. All the calculations are carried out using the freely available routine PKQuest Java (http://www.pkquest.com) which has an easy to use, simple interface.ResultsTheoretically, the AM permeability provides an accurate estimate of the intestinal DC permeability for solutes whose absorption ranges from 1% to 99%. The experimental human AM permeabilities determined by deconvolution are similar to those determined by direct human jejunal perfusion. The small intestinal pH varies with position and the results are interpreted in terms of the pH dependent octanol partition. The permeability versus partition relations are presented separately for the uncharged, basic, acidic and charged solutes. The small uncharged solutes caffeine, acetaminophen and antipyrine have very high permeabilities (about 20 x 10-4 cm/sec) corresponding to an unstirred layer of only 45 μm. The weak acid aspirin also has a large AM permeability despite its low octanol partition at pH 7.4, suggesting that it is nearly completely absorbed in the first part of the intestine where the pH is about 5.4.ConclusionsThe AM deconvolution method provides an accurate estimate of the human intestinal permeability. The results for these 90 drugs should provide a useful benchmark for evaluating QSAR models.

Beyond Lebesgue and Baire III: Steinhausʼ Theorem and its descendants

We generalize from the real line to normed groups (i.e., the invariantly metrizable, right topological groups) a result on infinite combinatorics, valid for ‘large’ subsets in both the category and measure senses, which implies both Steinhausʼs Theorem and many of its descendants. We deduce the inherent measure-category duality in this setting directly from properties of analytic sets. We apply the result to extend the Pettis Theorem and the Continuous Homomorphism Theorem to normed groups, i.e., beyond metrizable topological groups, and in a companion paper deduce automatic joint-continuity results for group operations.

Application of the maximal entropy production principle to rapid solidification: A multi-phase-field model

Because the Cahn–Hillard and Allen–Cahn equations cannot deal with the additional constraints in the multi-phase-field models, several approximate treatments, e.g. a specific partition relation and the condition of equal or unequal diffusion potentials, were proposed. In this paper, the problem is solved successfully by the maximal entropy production principle and a model is developed for rapid solidification of a binary alloy system. Due to the mixture law used to define the free energy density, solute concentration and chemical potential jumps happen at an “imaginary” sharp interface between solid and liquid. The solute diffusions in solid and liquid are described by two independent equations and additional non-linear equations do not need to be employed to fix the solute concentrations of solid and liquid. Application to solute trapping during rapid solidification of Si–9at.% As alloy shows that a good agreement between the model predictions and the experimental results is obtained. The interface and bulk contributions are decoupled at very low and very high interface velocities and in other cases the interaction between them depends weakly on the interface velocity.

Calibration of an Empirical Thermometer and Oxybarometer based on the Partitioning of Sc, Y and V between Olivine and Silicate Melt

The evolution of basaltic magmas depends on their redox state, hence oxygen fugacity, but there is increasing evidence that this intensive thermodynamic variable may be less well understood in basalts than commonly supposed. The redox state of terrestrial basalts has to a large extent been inferred from the Fe3+/Fe2+ ratios of their quenched glasses. However, this quantity appears to be significantly affected during late and post-eruptive processes in magmatic systems (e.g. by degassing, charge-transfer reactions of redox-variable species, and alteration), so that the degree to which the Fe3+/Fe2+ ratios preserved in basaltic glasses reflect the oxidation state of the magma at high temperature is unclear. Because olivine is the first silicate mineral to crystallize from primitive basaltic liquids on cooling following decompression, the equilibrium partitioning relations preserved in olivine phenocrysts in basalts are, in principle, less disturbed by these late and post-eruptive processes and, therefore, may better reflect the high-temperature (pre-eruptive) conditions of the magma. Here we calibrate an oxybarometer based on the strong sensitivity of the partitioning of vanadium between olivine and silicate melt to oxygen fugacity. Our empirical parameterization, calibrated over a range of redox conditions between four log(10) units above and below the quartz-fayalite-magnetite (QFM) oxygen buffer, takes into account the effects of temperature, olivine composition (i.e. Mg/Fe ratios) and melt composition (namely the activities of CaO, SiO2, AlO1 center dot 5, NaO0 center dot 5 and KO0 center dot 5), and allows oxygen fugacity determinations to within similar to 0 center dot 25 log(10) units. We also explore the sensitivity of the exchange partitioning of Sc and Y between olivine and melt to temperature as a geothermometer. Our calibration indicates that this geothermometer allows temperature to be estimated to within 15 degrees C, but precision is strongly dependent on the Sc and Y measurements in olivine and melt.

Socle chains of abelian regular semiartinian rings

Let R be an abelian regular and semiartinian ring with socle chain (Sα∣α≤σ). If λα denotes the rank of the semisimple module Sα+1/Sα, for every α<σ, then the dimension sequence(λα∣α<σ) is an invariant for R. By applying classical results of combinatorial set theory we prove necessary conditions satisfied by this invariant. On the other hand, we present constructions of commutative regular semiartinian rings with given ranks of slices of socle chain. In some particular cases we prove a necessary and sufficient condition under which there exists an abelian regular semiartinian ring with given ranks of slices.

Combinatorial Aspects of the Splitting Number

This paper deals with the splitting number \({\mathfrak{s}}\) and polarized partition relations. In the first section we define the notion of strong splitting families, and prove that its existence is equivalent to the failure of the polarized relation$$\left(\begin{array}{lll}\mathfrak{s} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}\mathfrak{s} \\ \omega \end{array} \right)}^{1, 1}_{2}$$. We show that the existence of a strong splitting family is consistent with ZFC, and that the strong splitting number equals the splitting number, when it exists. Consequently, we can put some restriction on the possibility that s is singular. In the second section we deal with the polarized relation under the weak diamond, and we prove that the strong polarized relation$$\left(\begin{array}{lll}2^{\omega} \\ \omega \end{array} \right) \rightarrow {\left(\begin{array}{ll}2^{\omega} \\ \omega \end{array} \right)}^{1, 1}_{2}$$is consistent with ZFC, even when cf \({(2^{\omega}) = \aleph_{1}}\) (hence the weak diamond holds).

Transforming rectangles into squares, with applications to strong colorings

It is proved that every singular cardinal λ admits a function rts:[λ+]2→[λ+]2 that transforms rectangles into squares. Namely, for every cofinal subsets A,B of λ+, there exists a cofinal subset C⊆λ+, such that rts[A⊛B]⊇C⊛C.As a corollary, we get that for every uncountable cardinal λ, the classical negative partition relation λ+⁄→[λ+]λ+2 coincides with the following syntactically stronger statement. There exists a function f:[λ+]2→λ+ such that for every positive integer n, every family A⊆[λ+]n of size λ+ of mutually disjoint sets, and every coloring d:n×n→λ+, there exist a,b∈A with max(a)<min(b) such that c(ai,bj)=d(i,j) for all i,j<n.