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Expanding on my comment and continuing GH from MO's answer: For $\alpha >2$, let $r:=\lceil\alpha/2\rceil+1$. Since a half-open interval $I$ of length $\alpha$ cannot contain $r$ numbers whose pairwi …

The four examples can be built by the following four operations on set systems, starting from empty systems on one-element set systems of type $(\{x\},\{\emptyset\})$: I) subsystem: If $(A,H)$ is a s …

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- …

regarding Question 2: By Theorem 8.4.3 in M. Lothaire, Applied Combinatorics on Words and subsequent discussion, the number of $k$th powers in $S$ can be determined in time $O(N)$. An algorithm for …

Q1: yes, this is a theorem by Wilson; see the first paragraph here: https://arxiv.org/abs/1604.07282 Edit: perhaps the book Decomposition of graphs by J. Bosak might be helpful (the preview on google …

A lower bound $A_m\ge (\Omega(m))!$ can be obtained as follows. Let $n$ be a positive integer and $\pi$ an arbitrary permutation of $\{1,2,\dots,n\}$. Construct a graph $G(\pi)=(V,E(\pi))$ as follows. …

By domotorp's answer https://mathoverflow.net/q/362229 to a previous question, De Bruijn-Erdos theorem characterizes complete linear collections as near-pencils or finite projective planes. The image …

Let $S=\Sigma v_i$. If $S=0$, sort the vectors according to their angle along the unit circle. Then the corresponding closed path traces the boundary of a convex polygon. In fact, the vectors $v_i$ ca …

For $n=5$, this has been shown by Eppstein: D. Eppstein, Happy endings for flip graphs, Journal of Computational Geometry 1 (2010), no. 1, 3--28. For odd $n>5$, one could consider the polygon bounded …

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 question (1), $k$ grows linearly with $n$, so it cannot be chosen independently. Color the vertices of the grid black and white, so that the vertices $(x,y)$ with $x+y$ even are black and the rem …

There has been some research on a related problem, finding a smallest transversal for sets of rectangles with a given maximum number of disjoint rectangles. Small transversal implies many rectangles i …

The first question essentially asks what is the maximum chromatic number of the minimum distance graph of a finite set of points in the plane. Let $S$ be a planar point set and let $m$ be the minimum …

Graphs obtained from a (directed) cycle by a repeated operation of attaching a (directed) cycle to a vertex of degree $2$ have unique Eulerian tour. The sequence appears in OEIS: http://oeis.org/A1026 …

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 …