Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

A classification of separable Rosenthal compacta and its applications

The present work consists of three parts. In the first one we determine the prototypes of separable Rosenthal compacta and we provide a classification theorem. The second part concerns an extension of a theorem of S. Todorcevic. The last one is devoted to applications.Comment: 55 pages, no figure

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known

Chains, Antichains, and Complements in Infinite Partition Lattices

We consider the partition lattice $\Pi_\kappa$ on any set of transfinite cardinality $\kappa$ and properties of $\Pi_\kappa$ whose analogues do not hold for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the cardinality of any maximal well-ordered chain is always exactly $\kappa$; (II) there are maximal chains in $\Pi_\kappa$ of cardinality $> \kappa$; (III) if, for every cardinal $\lambda < \kappa$, we have $2^{\lambda} < 2^\kappa$, there exists a maximal chain of cardinality $< 2^{\kappa}$ (but $\ge \kappa$) in $\Pi_{2^\kappa}$; (IV) every non-trivial maximal antichain in $\Pi_\kappa$ has cardinality between $\kappa$ and $2^{\kappa}$, and these bounds are realized. Moreover we can construct maximal antichains of cardinality $\max(\kappa, 2^{\lambda})$ for any $\lambda \le \kappa$; (V) all cardinals of the form $\kappa^\lambda$ with $0 \le \lambda \le \kappa$ occur as the number of complements to some partition $\mathcal{P} \in \Pi_\kappa$, and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition; (VI) Under the Generalized Continuum Hypothesis, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201

An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games

An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in ${\mathbb R}^n$. This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u

On bisequentiality and spaces of strictly decreasing functions on trees

We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of "A filter on a collection of finite sets and Eberlein compacta" by T. Cie\'sla. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta.Comment: 9 page