Partition Relations Research Articles

Partitioning subsets of generalised scattered orders

In 1956, 48 years after Hausdorff provided a comprehensive account on ordered sets and defined the notion of a scattered order, Erdős and Rado founded the partition calculus in a seminal paper. The present paper gives an account of investigations into generalisations of scattered linear orders and their partition relations for both singletons and pairs. We consider analogues for these order-types of known partition theorems for ordinals or scattered orders and prove a partition theorem from assumptions about cardinal characteristics. Together, this continues older research by Erdős, Galvin, Hajnal, Larson and Takahashi and more recent investigations by Abraham, Bonnet, Cummings, Džamonja, Komjáth, Shelah and Thompson.

Topological partition relations for countable ordinals

Topological partition relations for countable ordinals

INFINITE COMBINATORICS PLAIN AND SIMPLE

AbstractWe explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

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.

Suslin trees, the bounding number, and partition relations

We investigate the unbalanced ordinary partition relations of the form λ → (λ, α)2 for various values of the cardinal λ and the ordinal α. For example, we show that for every infinite cardinal κ, the existence of a κ+-Suslin tree implies κ+ ↛ (κ+, log κ (κ+) + 2)2. The consistency of the positive partition relation b → (b, α)2 for all α < ω1 for the bounding number b is also established from large cardinals.

Strong sequences and partition relations

Abstract The first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

THE HALPERN–LÄUCHLI THEOREM AT A MEASURABLE CARDINAL

AbstractSeveral variants of the Halpern–Läuchli Theorem for trees of uncountable height are investigated. Forκweakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finited≥ 2, we prove the consistency of the Halpern–Läuchli Theorem ondmany normalκ-trees at a measurable cardinalκ, given the consistency of aκ+d-strong cardinal. This follows from a more general consistency result at measurableκ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

A review of phosphorus partition relations for use in basic oxygen steelmaking

Phosphorus removal in basic oxygen steelmaking is a significant problem for integrated steelmakers. Phosphorus removal is required due to its deleterious effect on the mechanical properties of steel. However, this is progressively becoming more difficult due to the increasing phosphorus content of many iron ores. Many studies have investigated dephosphorisation and published empirical phosphorus partition (LP) equations for a range of conditions. The structure of these equations has been used to develop a new partition relation that allows the effect of minor slag constituents such as TiO2, Al2O3 and V2O5 on steel dephosphorisation to be tested. Al2O3 was found to have a weak negative effect on the measured LP, except at the lower oxygen potential range tested, where a positive correlation was observed. Increasing TiO2 and V2O5 contents were found to decrease the measured LP; however, these correlations became less prevalent at the higher oxygen potential ranges tested.

Bayesian Poisson calculus for latent feature modeling via generalized Indian Buffet Process priors

Statistical latent feature models, such as latent factor models, are models where each observation is associated with a vector of latent features. A general problem is how to select the number/types of features, and related quantities. In Bayesian statistical machine learning, one seeks (nonparametric) models where one can learn such quantities in the presence of observed data. The Indian Buffet Process (IBP), devised by Griffiths and Ghahramani (2005), generates a (sparse) latent binary matrix with columns representing a potentially unbounded number of features and where each row corresponds to an individual or object. Its generative scheme is cast in terms of customers entering sequentially an Indian Buffet restaurant and selecting previously sampled dishes as well as new dishes. Dishes correspond to latent features shared by individuals. The IBP has been applied to a wide range of statistical problems. Recent works have demonstrated the utility of generalizations to nonbinary matrices. The purpose of this work is to describe a unified mechanism for construction, Bayesian analysis, and practical sampling of broad generalizations of the IBP that generate (sparse) matrices with general entries. An adaptation of the Poisson partition calculus is employed to handle the complexities, including combinatorial aspects, of these models. Our work reveals a spike and slab characterization, and also presents a general framework for multivariate extensions. We close by highlighting a multivariate IBP with condiments, and the role of a stable-Beta Dirichlet multivariate prior.

In memoriam: James Earl Baumgartner (1943–2011)

James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations.

A diagnostic approach to constraining flow partitioning in hydrologic models using a multiobjective optimization framework

AbstractHydrologic models are often tasked with replicating historical hydrographs but may do so without accurately reproducing the internal hydrological functioning of the watershed, including the flow partitioning, which is critical for predicting solute movement through the catchment. Here we propose a novel partitioning‐focused calibration technique that utilizes flow‐partitioning coefficients developed based on the pioneering work of L'vovich (1979). Our hypothesis is that inclusion of the L'vovich partitioning relations in calibration increases model consistency and parameter identifiability and leads to superior model performance with respect to flow partitioning than using traditional hydrological signatures (e.g., flow duration curve indices) alone. The L'vovich approach partitions the annual precipitation into four components (quick flow, soil wetting, slow flow, and evapotranspiration) and has been shown to work across a range of climatic and landscape settings. A new diagnostic multicriteria model calibration methodology is proposed that first quantifies four calibration measures for watershed functions based on the L'vovich theory, and then utilizes them as calibration criteria. The proposed approach is compared with a traditional hydrologic signature‐based calibration for two conceptual bucket models. Results reveal that the proposed approach not only improves flow partitioning in the model compared to signature‐based calibration but is also capable of diagnosing flow‐partitioning inaccuracy and suggesting relevant model improvements. Furthermore, the proposed partitioning‐based calibration approach is shown to increase parameter identifiability. This model calibration approach can be readily applied to other models.

Choiceless Ramsey Theory of Linear Orders

Motivated by work of Erdős, Milner and Rado, we investigate symmetric and asymmetric partition relations for linear orders without the axiom of choice. The relations state the existence of a subset in one of finitely many given order types that is homogeneous for a given colouring of the finite subsets of a fixed size of a linear order. We mainly study the linear orders 〈α2,<lex〉, where α is an infinite ordinal and <lex is the lexicographical order. We first obtain the consistency of several partition relations that are incompatible with the axiom of choice. For instance we derive partition relations for 〈ω2,<lex〉 from the property of Baire for all subsets of ω2 and show that the relation \(\langle ^{\kappa }{2}, <_{lex}\rangle \longrightarrow (\langle ^{\kappa }{2}, <_{lex}\rangle )^{2}_{2}\) is consistent for uncountable regular cardinals κ with κ<κ = κ. We then prove a series of negative partition relations with finite exponents for the linear orders 〈α2,<lex〉. We combine the positive and negative results to completely classify which of the partition relations \(\langle ^{\omega }{2}, <_{lex}\rangle \longrightarrow (\bigvee _{\nu <\lambda }K_{\nu },\bigvee _{\nu <\mu }M_{\nu })^{m}\) for linear orders Kν,Mν and m≤4 and 〈ω2,<lex〉→(K,M)n for linear orders K,M and natural numbers n are consistent.

Partitions and conservativity

We study the partition properties enjoyed by the “next best thing to a P-point” ultrafilters introduced recently in joint work with Dobrinen and Raghavan. That work established some finite-exponent partition relations, and we now analyze the connections between these relations for different exponents and the notion of conservativity introduced much earlier by Phillips. In addition, we establish some infinite-exponent partition relations for these ultrafilters and also for sums of non-isomorphic selective ultrafilters indexed by selective ultrafilters.

First-order combinatorics and model-theoretical properties that can be distinct for mutually interpretable theories

The notions of finitary and infinitary combinatorics were recently introduced by the author. In the present article, we discuss these notions and the corresponding semantical layers. We suggest a definition of a model-theoretical property. By author’s opinion, this definition agrees with the meaning that is generally accepted and used inmodel theory.We show that the similarity relation for theories over finitary and infinitary layers of model-theoretical properties is natural and important. Our arguments are based on comparing our approach with known model-theoretical ones.We find examples of pairs of mutually interpretable theories possessing distinct simple model-theoretical properties. These examples show weak points of the notion of mutual interpretability from the point of view of preservation of model-theoretical properties.

THE TOPOLOGICAL PIGEONHOLE PRINCIPLE FOR ORDINALS

AbstractGiven 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.

Open and solved problems concerning polarized partition relations

We list some open problems, concerning the polarized partition relation. We solve a couple of them by showing that for every singular cardinal $\mu$ one can force the strong polarized relation with respect to the pair $\mu^+,\mu$.

THE COUNTERPARTS TO STATEMENTS THAT ARE EQUIVALENT TO THE CONTINUUM HYPOTHESIS

AbstractWe consider natural ${\rm{\Sigma }}_2^1$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these ${\rm{\Sigma }}_2^1$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for ${\rm{\Sigma }}_2^1$ colourings which hold precisely when there is a non-constructible real.

Subtlety and partition relations

We study the subtlety of a cardinal κ and of . We show that, under a certain large cardinal assumption, it is consistent that is subtle for some but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of can be characterized by a certain partition relation on . We also study the property of faintness of κ, and the subtlety of with the strong inclusion.

A transformation scheme for infinitary first-order combinatorics presenting computational level of expressiveness in predicate logic

We continue to develop a new first-order combinatorial approach presenting a conceptual framework for investigations concerning expressive power of first-order logic. In this work, we consider the case of infinitary first-order combinatorics. Based on the universal construction of finitely axiomatizable theories, we introduce some common scheme yielding finitely axiomatizable theories with pre-assigned sets of model-theoretic properties. At an initial stage, a maximum common Turing’s computation is performed (one can say, computable Brute Force). Starting from an input block of parameters, the computation yields a computably axiomatizable theory T. Finally, by applying an available version of the universal construction, the theory T is transformed into a finitely axiomatizable theory F that inherits model-theoretic properties of T within the infinitary semantic layer. We also give three demonstrations showing possibilities of this method.

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]&lt;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.