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Yes, there are non-square rectangles that admit a perfect squaring. The smallest number of squares in a perfect squaring of a rectangle is 9. On the other hand the smallest number of squares in a per …

Update. The recent paper Every $d(d+1)$-connected graph is globally rigid in $\mathbb{R}^d$ by Soma Villányi gives a positive answer to the question. Old Answer. I think this is still an open problem, …

Pilipczuk and Siebertz proved that every planar graph has such a partition with an even stronger property. Namely, each part $V_i$ is a geodesic path, and the graph obtained by contracting each part …

In a comment to this question, Tim Gowers remarked that using the axiom of choice, one can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet to …

Regarding III, the Alexandrov-Urysohn Theorem gives sufficient conditions. Any zero-dimensional, separable, nowhere compact, and completely metrizable space is homeomorphic to $J$.

In the case that the pursuers have to actually catch the fugitive, this was answered in the article Escaping an infinitude of lions by Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff …

No polynomial-time algorithm exists, unless P=NP. Indeed, even for TSP instances where all distances are $1$ or $2$ (note that these automatically satisfy the triangle inequality), Engebretsen and Kar …

For A), here is a construction that gives $2\binom{n}{3}$ defective triples, which is almost best possible. Let $D$ be a digraph with vertex set $[n]$ and arcs $(i,i+1)$ and $(i+1, i)$ for all $i \in …

There is a wide body of work on this in connection with the classic De Bruijn–Erdős theorem. De Bruijn–Erdős Theorem. Every set of $n$ points in the plane (not all lying on the same line) deter …

The answer to the third question is no. This is a rather counter-intuitive discovery of Micha Perles from the sixties. See this paper of Ziegler, for a simpler construction and other pertinent infor …

This sort of addresses the question in your last paragraph, but it is actually a bit tangential. Hopefully it is still of interest. It is well known that every planar graph has an embedding such t …

No. The set of all $\{0,1\}$-sequences is also a clique in $G$. Thus, $\chi(G) \geq 2^{\aleph_0}$. On the other hand, the set of all bounded real sequences has size $2^{\aleph_0}$, so $\chi(G)=2^{\ …

Here is a generalization to arbitrary finite metric spaces. Recall that the Sylvester-Gallai theorem easily implies the following theorem. Theorem to be generalized. Every non-collinear set of $n$ …

This is an answer to your last question. As far as I know, it is still open whether there exists a dense subset $S$ of the plane with all pairwise distances rational. Such a set $S$ would imply a po …

Yes. See this paper of Negami. The main result is that for any fixed knot (or link) of type $k$, there is a constant $R(k)$ such that every straight line embedding of $K_{R(k)}$ in $\mathbb{R}^3$ con …