Related papers: Dense free sets
We prove the decidability of the elementary theory of a free group.
An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…
We show that the problem of counting perfect matchings remains #P-complete even if we restrict the input to very dense graphs, proving the conjecture in [5]. Here "dense graphs" refer to bipartite graphs of bipartite independence number…
A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In…
A topological space is iso-dense if it has a dense set of isolated points. A topological space is scattered if each of its non-empty subspaces has an isolated point. In $\mathbf{ZF}$, in the absence of the axiom of choice, basic properties…
We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…
We construct certain non-degenerate maps and sets, mainly in the complex-analytic category. For example, we show that for every countable subset S in an irreducible complex space X there exists a holomorphic map from the unit disk to X such…
In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.
We prove an interpolation theorem for bounded free holomorphic functions.
It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and…
We prove that for every prime $p$ algebraically clean graphs of groups are virtually residually $p$-finite and cohomologically $p$-complete. We also prove that they are cohomologically good. We apply this to certain $2$-dimensional Artin…
In this work we study the problem of existence of symplectic structures on free nilpotent Lie algebras. Necessary and sufficient conditions are given for even dimensional ones. The one dimensional central extension for odd dimensional free…
In this paper a generalized topological central point theorem is proved for maps of a simplex to finite-dimensional metric spaces. Similar generalizations of the Tverberg theorem are considered.
We prove that for a Baire-generic Riemannian metric on a closed smooth manifold, the union of the images of all stationary geodesic nets forms a dense set.
We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega$ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by…
We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a…
It is consistent with ZF set theory that the Euclidean topology on the real line is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres.
We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that…
In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the…
We characterize the subsets $E \subset \mathbb{R}$ for which there exists a continuous real valued function $f: \mathbb{R}\to\mathbb{R}$ such that lip $f$ is finite everywhere and Lip $f$ is infinite exactly on $E$.