Related papers: Pach's selection theorem does not admit a topologi…

We discuss the class of paratopological groups which admits a transversal, $T_{1}$-independent and $T_{1}$-complementary paratopological group topology. We show that the Sorgenfrey line does not admit a $T_{1}$-complementary Hausdorff…

Given finite sets $X_1,\dotsc,X_m$ in $\mathbb{R}^d$ (with $d$ fixed), we prove that there are respective subsets $Y_1,\dotsc,Y_m$ with $|Y_i|\ge \frac{1}{\operatorname{poly}(m)}|X_i|$ such that, for $y_1\in Y_1,\dotsc,y_m\in Y_m$, the…

The Upper Bound Theorem for convex polytopes implies that the $p$-th Betti number of the \v{C}ech complex of any set of $N$ points in $\mathbb R^d$ and any radius satisfies $\beta_{p} = O(N^{m})$, with $m = \min \{ p+1, \lceil d/2 \rceil…

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Lof's type theory (hence…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of…

We prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally…

We study ultrafilters from the perspective of the algebra in the \v{C}ech-Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp. a…

In this note, as a particular case of a more general result, we obtain the following theorem: Let $\Omega\subseteq {\bf R}^n$ be a non-empty bounded open set and let $f:\overline {\Omega}\to {\bf R}^n$ be a continuous function which is…

This paper introduces a new problem concerning additive properties of convex sets. Let $S= \{s_1 < \dots <s_n \}$ be a set of real numbers and let $D_i(S)= \{s_x-s_y: 1 \leq x-y \leq i\}$. We expect that $D_i(S)$ is large, with respect to…

We study the problem of augmenting the locus $\mathcal{N}_{\ell}$ of a plane Euclidean network $\mathcal{N}$ by inserting iteratively a finite set of segments, called \emph{shortcut set}, while reducing the diameter of the locus of the…

It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the…

The $\text{PSL}(4,\mathbb{R})$ Hitchin component of a closed surface group $\pi_1(S)$ consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of $S$. We prove that the leaves of the…

Let $S(\phi)= \{z:\;|\arg(z)|\geq \phi\}$ be a sector on the complex plane $\CC$. If $\phi\geq \pi/2$, then $S(\phi)$ is a convex set and, according to the Gauss-Lucas theorem, if a polynomial $p(z)$ has all its zeros on $S(\phi)$, then the…

Let $P_1, P_2,\ldots, P_{d+1}$ be pairwise disjoint $n$-element point sets in general position in $d$-space. It is shown that there exist a point $O$ and suitable subsets $Q_i\subseteq P_i \; (i=1, 2, \ldots, d+1)$ such that $|Q_i|\geq…

We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional…

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field $k$. Let $H=kG(\chi, a,\d)$ be a Hopf-Ore extension of $kG$ and $H'$ a rank one…

Vogel's universality implies a unified description of the adjoint sector of representation theory for simple Lie algebras in terms of three parameters $\alpha,\beta,\gamma$, which are homogeneous coordinates of Vogel's plane. Actually this…

Under $\mathrm{ZF}$, we show that the statement that every subset of every $\mathbb{R}$-vector space has a maximal convex subset is equivalent to the Axiom of Choice. We also study the strength of the same statement restricted to some…

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence…