Finite-dimensional objects in distinguished triangles

Equilateral Triangles in Finite

Some finite groups with divisibility graph containing no triangles

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...

Equilateral Triangles in Finite Metric Spaces

In the context of finite metric spaces with integer distances, we investigate the new Ramsey-type question of how many points can a space contain and yet be free of equilateral triangles. In particular, for finite metric spaces with distances in the set {1, . . . , n}, the number Dn is defined as the least number of points the space must contain in order to be sure that there will be an equilat...

Lines Tangent to Four Triangles in Three-Dimensional Space

We investigate the lines tangent to four triangles in R3. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even.

Large Triangles in the d-Dimensional Unit-Cube

We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 for a distribution of n points in the d-dimensional unit-cube [0, 1] such that the minimum (2-dimensional) area of a triangle among these n points is maximal. Denoting this maximum value by Δoff−line d (n) and Δ on−line d (n) for the off-line and the online situation, respectively, we will show that c1 ·(l...