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Similar questions are studied by Martin Balko and Pavel Valtr in their recent paper: http://www.sciencedirect.com/science/article/pii/S0195669817300847 They use SAT solvers to obtain counterexamples …

By coloring the Horton set with two colors, periodically mod 3 according to the $x$-coordinate, Devillers et al. obtained arbitrarily large bicolored point sets with no monochromatic empty convex $5$- …

Actually, for any finite subset $P$ of the plane, there are $2^{\aleph_0}$ monochromatic scaled copies of $P$: https://arxiv.org/abs/1304.3154 (see also the question Monochromatic point sets in two- …

The answer to the second question is yes. For $n=2^m-1$, there exists a binary Hamming code, which is a special type of a linear code. It is a linear subspace $H$ of $\mathbb{Z}_2^n$ of dimension $n- …

This seems to be an open problem: Rom Pinchasi, On the odd area of the unit disc, Israel Journal of Mathematics 256, 619-637. https://doi.org/10.1007/s11856-023-2518-4 Amir Carmel and Rom Pinchasi, So …

For every set $A$ of six points in general position (no three on a line) there is at most one such partition, which can be seen as follows. First, if three segments determined by $A$ cross at the sa …

The number $N_{6,3}$ does not exist: let $V_1=\mathrm{span}(e_1,e_2,e_3)$, $V_2=\mathrm{span}(e_3,e_4,e_5)$, $V_3=\mathrm{span}(e_1,e_6,e_5)$, and $V_j=\mathrm{span}(e_1,e_3, e_4+j\cdot e_5)$ for $j\g …

Theorem 5 in [9] proves that a simple closed curve with arbitrary metric contains vertices of an equilateral triangle. In particular, this applies to any simple closed curve in $\mathbb{R}^n$.

Already in the plane there are arrangements such that the longest path in their dual covers roughly at most $2/3$ of the regions, therefore they are not Hamiltonian. The reason is that the dual graph …

A family of so-called Hadwiger-Hill simplices in $\mathbb{R}^d$ (https://en.wikipedia.org/wiki/Hill_tetrahedron) can be cut into $m^d$ congruent simplices, all similar to the original simplex; that is …

This is an approximation of the answer. The main message is that, indeed, the probability of disconnection is dominated by isolated vertices. I will assume that $\delta=1/m$, for some large positive …

Q2: If the union of the discs is conected, or more generaly, if no line separates the discs in a nontrivial way, then the integral of the lengths of the projections over $[0, \pi)$ is equal to the per …

Sometimes it is better to place the ball "in the corner". For example: Let $n=8$ and $R=\sqrt{\frac{7}{8}}$. Then the ball with center $(\frac{1}{8}, \frac{1}{8}, \dots, \frac{1}{8})$ and radius $R$ …

Regarding Q1: The graph is a subgraph of the visibility graph of the integer lattice. Every sublattice $x+2\mathbb{Z} \times 2\mathbb{Z}$ is an independent set in the visibility graph, and the integer …

The largest number is $g(x,d) = (x+2)^d$, by an analogous argument as for visibility: if the differences of all coordinates of two points $A,B$ are divisible by $x+2$, then there are at least $x$ latt …