Related papers: Filter-linkedness and its effect on preservation o…
We consider profunctors $f : P \promap Q$ between posets and introduce their {\em graph} and {\em ascent}. The profunctors $\Pro(P,Q)$ form themselves a poset, and we consider a partition $\cI \sqcup \cF$ of this into a down-set $\cI$ and…
Let $\mathcal{C}$ be a small, connected category with finite hom-sets. We show that if the embedding of a connected subcategory $\mathcal{J}$ is both initial and final, then the restriction of any $\mathcal{C}$-module along $\mathcal{J}$…
We investigate large set axioms defined in terms of elementary embeddings over constructive set theories, focusing on $\mathsf{IKP}$ and $\mathsf{CZF}$. Most previously studied large set axioms, notably the constructive analogues of large…
Motivated by questions about simplification of topology, we take a discrete approach to the dependency of simplifying operations, using methods based on combinatorial gradient dynamics. We interpret the filter in persistent homology as a…
Denote by $\continuum=2^{\aleph_0}$ the cardinal of continuum. We construct an intriguing family $(P_\alpha: \alpha\in\continuum)$ of prime $z$-ideals in $\C_0(\reals)$ with the following properties: If $f\in P_{i_0}$ for some…
The union-closed sets conjecture states that in any nonempty union-closed family $\mathcal{F}$ of subsets of a finite set, there exists an element contained in at least a proportion $1/2$ of the sets of $\mathcal{F}$. Using the…
The Fon-Der-Flaass action partitions the order ideals of a poset into disjoint orbits. For a product of two chains, Propp and Roby observed --- across orbits --- the mean cardinality of the order ideals within an orbit to be invariant. That…
Given a union-closed family $\mathcal{F}$ of subsets of the universe $[n]$, with $\mathcal{F}$ not equal to the power set of $[n]$, a new subset $A$ can be added to it such that the resulting family remains union-closed. We construct a new…
We show that the $f$-vector of Galashin's poset associahedron $\mathscr A(P)$ only depends on the comparability graph of $P$. In particular, this allows us to produce a family of polytopes with the same $f$-vectors as permutohedra, but that…
The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in the present work and for updating the…
A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further…
We use known finite support iteration techniques to present various examples of models where several cardinal characteristics of Cicho\'n's diagram are pairwise different. We show some simple examples forcing the left-hand side of…
A set of reals $X$ is $\mathfrak{b}$-concentrated if it has cardinality at least $\mathfrak{b}$ and it contains a countable set $D\subseteq X$ such that each closed subset of $X$ disjoint with $D$ has size smaller than $\mathfrak{b}$. We…
Selective ultrafilters are characterized by many equivalent properties, in particular the Ramsey property that every finite colouring of unordered pairs of integers has a homogeneous set in U, and the equivalent property that every function…
We introduce more properties of forcing notions which imply that their lambda-support iterations are lambda-proper, where lambda is an inaccessible cardinal. This paper is a direct continuation of section A.2 of math.LO/0210205. As an…
In this note, given a family of relative dimension one over a smooth curve, we determine the parity of the restriction of a relative theta characteristic to an arbitrary multiple of a fiber in terms of the parity of the restriction to a…
We prove that the generic maximal independent family obtained by iteratively forcing with the Mathias forcing relative to diagonalization filters is densely maximal. Moreover, by choosing the filters with some care one can ensure the family…
A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in…
We isolate a new class of ultrafilters on N, called "quasi-selective" because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of…
Assume $X$ is a variety over $\mathbb{C}$, $A \subseteq \mathbb{C}$ is a finitely generated $\mathbb{Z}$-algebra and $X_A$ a model of $X$ (i.e. $X_A \times_A \mathbb{C} \cong X$). Assuming the weak ordinarity conjecture we show that there…