Related papers: Chains of Baire class 1 functions and various noti…
In the 1970s M. Laczkovich posed the following problem: Let $\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an uncountable Polish space $X$ equipped with the pointwise ordering. \[\text{Characterize the order types…
Let $X$ be a Polish space and $K$ a separable compact subset of the first Baire class on $X$. For every sequence $\bs$ dense in $\kk$, the descriptive set-theoretic properties of the set \[ \lbf=\{L\in[\nn]: (f_n)_{n\in L} \text{is…
We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's…
We consider the set of Baire 1 functions endowed with the pointwise partial ordering and investigate the structure of the linearly ordered subsets.
We show that under the proper forcing axiom the class of all Aronszajn lines behave like $\sigma$-scattered orders under the embeddability relation. In particular, we are able to show that the class of better quasi order labeled fragmented…
A linear order $A$ is called strongly surjective if for every non empty suborder $B \preceq A$, there is an epimorphism from $A$ onto $B$ (denoted by $B \trianglelefteq A$). We show, answering some questions of D\'aniel T. Soukup, that…
Given a nonempty set $\mathcal{L}$ of linear orders, we say that the linear order $L$ is $\mathcal{L}$-convex embeddable into the linear order $L'$ if it is possible to partition $L$ into convex sets indexed by some element of $\mathcal{L}$…
In this article we provide an intrinsic characterization of the famous Howard-Bachmann ordinal in terms of a natural well-partial-ordering by showing that this ordinal can be realized as a maximal order type of a class of generalized trees…
We analyse the complexity of the class of (special) Aronszajn, Suslin and Kurepa trees in the projective hierarchy of the higher Baire-space $\omega_1^{\omega_1}$. First, we will show that none of these classes have the Baire property…
We prove that every weakly square compact cardinal is a strong limit cardinal. We also study Aronszajn trees with no uncountable finitely branching subtrees, characterizing them in terms of being Lindel\"of with respect to a particular…
In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non $\sigma$-scattered. This shows that a theorem of Laver, which…
The set of continuous or Baire class 1 functions defined on a metric space $X$ is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order)…
We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with…
Define the special tree number, denoted $\mathfrak{st}$, to be the least size of a tree of height $\omega_1$ which is neither special nor has a cofinal branch. This cardinal had previously been studied in the context of fragments of…
It is known that the set of permutations, under the pattern containment ordering, is not a partial well-order. Characterizing the partially well-ordered closed sets (equivalently: down sets or ideals) in this poset remains a wide-open…
We study discrete orderings in the real spectrum of a commutative ring by defining discrete prime cones and give an algebro-geometric meaning to some kind of diophantine problems over discretely ordered rings. Also for a discretely ordered…
We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings,…
It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of…
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
We consider a transitive relation on the power set of $\omega_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of…