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The line generated by two distinct points, $x$ and $y$, in a finite metric space $M=(V,d)$, denoted by $overline{xy}^M$, is the set of points given by $$overline{xy}^M:={zin V: d(x,y)=|d(x,z)pm d(z,y)|}.$$ A 2-set ${x,y}$ such that $overline{xy}^M=V$ is called a universal pair and its associated line a universal line. Chen and Chvatal conjectured that in any finite metric space either there is a universal line or there are at least $|V|$ different (non-universal) lines. Chvatal proved that this is indeed the case when the metric space has distances in the set ${0,1,2}$. Aboulker et al. proposed the following strengthening for Chen and Chvatal conjecture in the context of metric spaces induced by finite graphs. The number of lines plus the number of universal pairs is at least the number of point of the space. In this work we first prove that metric spaces with distances in the set ${0,1,2}$ satisfy this stronger conjecture. We also prove that for metric spaces induced by bipartite graphs the number of lines plus the number of bridges of the graph is at least the number its vertices, unless the graph is $C_4$ or $K_{2,3}$.
A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvatal conjectured that this holds for an arbitrary finite metric space, with a certain natural def
Let $mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $mathfrak{M}$-universal if every $Xinmathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions un
We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ Oleft(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + nright) $
Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Plucker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is emph{non-degenerate} with respect to $L$ (see belo
We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter