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By Theorem 1 from [DGM], for each $n\ge 9$, there exists a geometric thickness-two graph with $n$ vertices and $6n − 19$ edges. When we partition this graph into two straight-edged planar graphs and t …

Put $N=1$, $M=2$, $\Omega=\Bbb R^N$, and $u(x)=(x,0)$ for each $x\in\Bbb R^N$. Then the graph of $u$ is a straight line, so it has Hausdorff dimension $1=N$. On the other hand, let $C\subset [0,1]$ be …

I asked this question at MSE a week ago, but received no answer, so I cross-post it here. I obtained a negative answer to this MSE question provided each metric space $X$ such that $|X|=\frak c$ and d …

Is there a special name for a linearly ordered group $G$ such that for every positive element $g\in G$ there exists an element $h\in G$ such that $e<h<g$?

It is easy to show that $n$ (mutually different) circles on the plane can have maximum $n(n-1)$ intersection points. In our optimal graph drawing research we have encountered a counterpart of this res …

The original orchard-planting problem asks for the maximum number of $3$-point lines attainable by a configuration of points in the plane. I am interested in its natural generalization for (three-dime …

Clearly, there are examples for the second question. Each Hausdorff abelian paratopological group (that is, a group endowed with a topology making the multiplication continuous) which is not a topolog …

We are interested in the following Lemma. Let $V\subset [0,1]^n\subset\mathbb R^n$ be a set of $n$-dimensional vectors. Then for each $r\le |V|$ there exists a partition $$V=V_1\cup V_2\cup\dots \cup …

As I already told at the workshop, Taras Banakh and me solved this problem. But, surprisingly and converse to that I told at the workshop, the answer is affirmative. This is why is good to write a com …

I asked this question at MSE some months ago but I received only partial answers, so I put it here. The following sounds nice for me and I spent a good time during the investigation. But I am a profes …

I was involved into this subject when I answered this question from MSE. Trying to generalize my answer, I am thinking about a following Question. Let $X$ and $Y$ be metric spaces. When each continuo …

My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary …