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Mirna Džamonja

Publications and source records attributed to Mirna Džamonja.

At least 19 recordsLinked to original sources
arXiv2026

Iterating Generalised Perfect Set Forcing Along Well-Founded Orders

The technique of geometric forcing iteration was developed by Kanovei \cite{zbMATH01335192} and used to prove that the perfect set forcing can be iterated with countable supports along any partial order, while preserving $\aleph_1$. In \cite{Property-B} we considered a generalised perfect set forcing with respect to a filter on a cardinal $κ$ satisfying $κ^{<κ}=κ$, which we denoted ${\mathbb P} (\mathcal F)$, and we proved that its iteration with supports of size $\leκ$ along any ordinal preserves cardinals up to and including $κ^+$. We show that there is a version of the geometric iteration technique that applies to ${\mathbb P} (\mathcal F)$ and yields that for $κ$ satisfying $κ^{<κ}=κ$ and for appropriate filters $\mathcal F$, the forcing ${\mathbb P} (\FF)$ can be iterated with supports of size $\leκ$ along any well-founded partial order and preserve cardinals up to and including $κ^+$. As an application of our technique we obtain that common notions of arboreal forcings on $ω$ can be iterated with countable supports along any well-founded partial order and that such iterations preserve $\aleph_1$.

math.LO↗
arXiv2026

A note on iterating strongly $(<λ)$-closed stationary $λ^+$-cc forcing

We give an exposition of an iteration theorem for iterating $(<λ)$-closed stationary $λ^+$-cc forcing with supports of size $<λ$ and preserving these two properties. We discuss the relation of this theorem with other iteration theorems and forcing axioms that have appeared in the literature, notably the one from \cite{Sh80}.

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arXiv2026

Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$

We prove a theorem on iterated forcing that can be used for preservation of $\aleph_2$ and $\aleph_1$ in iterations with supports of size $\aleph_1$ of forcings that have amalgamation properties similar to those present in the perfect set forcing. The work is modelled after Baumgartner's Axiom A and his proof that iterations with countable support of the same preserve $\aleph_1$. In honour of James E. Baumgartner, the property introduced here is called Property B$(κ)$. The known additional difficulties when forcing at cardinals higher than $\aleph_1$ make for a less general theorem and a more complex theorem on the iteration, which is not an iteration theorem in the classical sense. The results extend to other cardinals $κ$ such that $κ^{<κ}=κ$, in place of $\aleph_1$. We give examples of individual forcings that have Property B$(κ)$ and their products. In particular, we introduce a correct version of the generalised perfect set forcing, which we call Perfect Set Forcing with Respect to a Filter. We give its basic properties and show that for the right kind of filter $\mathcal F$ this kind of forcing is iterable with supports of size $\leκ$.

math.LO↗
arXiv2026

Club guessing and the universal models

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element. The article was published in 2006. On rereading we noticed a missing parameter in Definition 1.1, which makes the rest the recounting of the Kojman-Shelah incorrect. We correct the (minor) error in this version, changes marked in red.

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arXiv2026

MSO logic of the real order with the set quantifiers ranging over the Borel sets

A celebrated 1969 theorem of Michael Rabin is that the MSO theory of the real order where the monadic quantifier is allowed only to range over the sets of rational numbers, is decidable. In 1975 Saharon Shelah proved that if the monadic quantifier is allowed to range over all subsets of the reals, the resulting MSO theory is undecidable. He conjectured that when we allow the monadic quantifier to range over the Borel subsets of the reals, the resulting MSO theory is decidable. We confirm this conjecture. Namely, the MSO theory of the real order where the set quantifier is allowed to range over the Borel sets, is decidable. If we only ask for the decidability in the language where each level of the Borel hierarchy is allowed a quantifier to denote sets of that level in the hierarchy, then we obtain a weaker MSO theory, which is not inly decidable but also interpretable in S2S.

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arXiv2025

Big Ramsey Degrees in Ultraproducts of Finite Structures

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $\fLL^*$ has, as a spine, $η_1$, an uncountable analogue of the order type of rationals $η$. Finite big Ramsey degrees for $η$ were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that $η_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to $η_1$ to show that it witnesses big Ramsey degrees of finite tuples in $η$ on every copy of $η$ in $η_1,$ and consequently in $\fLL^*$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

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arXiv2025

On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular, this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results. ERRATUM:We incorrectly claimed in Lemma 4.4(1) the formula $o(P\cdot Q)=o(P)\cdot o(Q)$ for wpos $P$ and $Q$ and incorrectly attributed it to Abraham and Bonnet. We incorrectly claimed in 4.4(2) that the formula for $h(P\cdot Q)$ was due to Abraham and Bonnet.

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arXiv2025

On the ABK Conjecture, alpha-well Quasi Orders and Dress-Schiffels product

The following is a 2008 conjecture of Abraham, Bonnet and Kubiś: [ABK Conjecture] Every well quasi order (wqo) is a countable union of better quasi orders (bqo). We obtain a partial progress on the conjecture, by showing that the class of orders that are a countable union of better quasi orders (sigma-bqo) is closed under various operations. These include diverse products, such as the Dress-Shieffels product. We develop various properties of the latter product. In relation with the main question, we explore the class of alpha-wqo for countable ordinals alpha and obtain several closure properties and a Hausdorff-style classification theorem. Our main contribution is the discovery of various properties of sigma-bqos and ruling out potential counterexamples to the ABK Conjecture.

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arXiv2025

On maximal order type of the lexicographic product

In the previously submitted version of this paper, available here for the record, we stated the following : "We give a self-contained proof of Isa Vialard's formula for $o(P\cdot Q)$ where $P$ and $Q$ are wpos. The proof introduces the notion of a cut of partial order, which might be of independent interest." In fact, the argument presented in the paper is wrong and Vialard formula has no known proof. I will try to prove the formula $o(P\cdot Q)=o(P)\cdot o(Q)$ from the [DzSS] paper because I believe that Altman's purported counter-example mentioned in the preprint is incorrect. This statement is written by Mirna Džamonja without consultation with Isa Vialard, who may hold different views. Mirna Džamonja has withdrawn her authorship from the conditionally accepted version of this note (IGPL) on January 20, 2025

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arXiv2022

On middle box products and paracompact cardinals

The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of many factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type ${}^{<κ}\square 2^λ$, where we prove that for a regular uncountable cardinal $κ$, if ${}^{<κ}\square 2^λ$ is paracompact for every $λ\geκ$, then $κ$ is at least inaccessible. The case of the products of the type ${}^{<κ}\square X^λ$ for $κ$ singular has not been studied much in the literature and we offer various results. The question if ${}^{<κ}\square 2^λ$ can be paracompact for all $λ$ when $κ$ is singular has been partially answered but remains open in general.

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arXiv2021

Chain Logic and Shelah's Infinitary Logic

For a cardinal of the form $κ=\beth_κ$, Shelah's logic $L^1_κ$ has a characterisation as the maximal logic above $\bigcup_{λ<κ} L_{λ, ω}$ satisfying Strong Undefinability of Well Order (SUDWO). SUDWO is a strengthening of the Undefinability of Well Order (UDWO). We prove that if $κ$ is singular of countable cofinality, Karp's chain logic \cite{Karpintroduceschain} is above $L^1_κ$, while it is already known that it satisfies UDWO and Interpolation. Moreover, we show that in these circumstances, the chain logic is -- in a sense -- maximal among logics with chain models to satisfy UDWO. We then show that the chain logic gives a partial solution to Problem 1.4. from Shelah's \cite{Sh797}, which asked whether for $κ$ singular of countable cofinality there was a logic strictly between $ L_{κ^+, ω}$ and $L_{κ^+, κ^+}$ having Interpolation. We show that modulo accepting as the upper bound a model class of $L_{κ, κ}$, Karp's chain logic satisfies the required properties. In addition, we show that this chain logic is not $κ$-compact, a question that we have asked on various occasions. We contribue to the further development of chain logic by proving the Union Lemma and identifying the chain-independent fragment of the logic, showing that it still has considerable expressive power. In conclusion, we have shown that the simply defined chain logic emulates the logic $L^1_κ$ in satisfying Interpolation, undefinability of well-order and maximality with respect to it, and the Union Lemma. In addition it has a Completeness Theorem.

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arXiv2021

Formalising Ordinal Partition Relations Using Isabelle/HOL

This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erdős--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all $m \in \mathbb{N}$, $ω^ω\arrows(ω^ω, m)$. This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.

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arXiv2020

Are all natural numbers the same

This is a report on state-of-the-art on the question of developing higher analogues of the forcing axiom PFA. Recently there have been several attempts to develop forcing axioms analogous to the proper forcing axiom (PFA) for cardinals of the form aleph_n where n > 1. We investigate the difficulties of doing this and survey some of the successes

math.LO↗
arXiv2020

On wide Aronszajn trees in the presence of MA

A wide Aronszajn tree is a tree of size and height $ω_1$ with no uncountable branches. We prove that under $MA(ω_1)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and Väänänen from 1994. We also prove that under the same assumption there is no universal Aronszajn tree, improving a result of Todorčevi{ć} from 2007 who proved the same under the assumption of BPFA for posets of size $\aleph_1$. Finally, we prove that under $MA(ω_1)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree.

math.LO↗
arXiv2020

Square Compactness and the filter extension property

We show that the consistency strength of $κ$ being $2^κ$-square compact is at least weak compact and strictly less than indescribable. This is the first known improvement to the upper bound of strong compactness obtained in 1973 by Hajnal and Juh{\' a}sz.

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