Related papers: Positional graphs and conditional structure of wea…
We answer several questions of V. Tka\v{c}uk from [Point-countable $\pi$-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55--69.] by showing that (1) there is a ZFC example of a first countable, 0-dimensional…
We improve previous work on the consistency strength of mutually stationary sequences of sets concentrating on points with divergent cofinality building on previous work by Adolf, Cox and Welch. Specifically, we have greatly reduced our…
We construct a model in which all $C$-sequences are trivial, yet there exists a $\kappa$-Souslin tree with full vanishing levels. This answers a question of Lambie-Hanson and Rinot, and provides an optimal combination of compactness and…
A set of sequences is said to converge simultaneously if there exists an infinite subset $H$ of the index set $\omega$ such that all sequences converge when restricted to $H$. We discuss simultaneous convergence of sequences in the same or…
Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary countably tight subspaces. The question if…
A topological space X$ has the Frechet-Urysohn property if for each subset A of X and each element x in the closure of A, there exists a countable sequence of elements of A which converges to x. Reznichenko introduced a natural…
We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every…
A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $\delta(G)=2$. We show that in…
We show that every Banach space in which weakly compact sets are super weakly compact in automatically weakly sequentially complete answering a question by Silber (2024). In the proof we show how to build a weakly compact set which is not…
If $(X,d)$ is a metric space then the map $f\colon X\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$, $x\neq y$. We determine the simplest non-closed sets $X\subseteq \mathbb{R}^n$ in the sense of…
We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most (2^{aleph_0})^V many levels of size omega. We also give a complete ZFC characterization of…
We give an algebraic characterization of pre-Hilbert spaces with an orthonormal basis. This characterization is used to show that there are pre-Hilbert spaces $X$ of dimension and density $\lambda$ for any uncountable $\lambda$ without any…
Let $G$ be an infinite compact group. We prove that for every cardinal $\kappa$ between the density and the weight of $G$, there exists a dense subgroup of $G$ of density $\kappa$.
A cornerstone theorem in the Graph Minors series of Robertson and Seymour is the result that every graph $G$ with no minor isomorphic to a fixed graph $H$ has a certain structure. The structure can then be exploited to deduce far-reaching…
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $\kappa$ is a singular strong limit cardinal and…
We characterize exactly the compactness properties of the product of \kappa\ copies of the space \omega\ with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard…
We prove the removable singularity theorem for nonlocal minimal graphs. Specifically, we show that any nonlocal minimal graph in $\Omega \setminus K$, where $\Omega \subset \mathbb{R}^n$ is an open set and $K \subset \Omega$ is a compact…
We investigate characterizations of uniformly rectifiable (UR) metric spaces by so-called weak Carleson conditions for flatness coefficients which measure the extent to which Hausdorff measure on the metric space differs from Hausdorff…
We show that a linearly ordered topological space is initially \lambda-compact if and only if it is \lambda-bounded, that is, every set of cardinality $\leq \lambda$ has compact closure. As a consequence, every product of initially…
A class of graphs is nowhere dense if for every integer r there is a finite upper bound on the size of cliques that occur as (topological) r-minors. We observe that this tameness notion from algorithmic graph theory is essentially the…