On the maximum number of isosceles right triangles in a finite point set
نویسندگان
چکیده
Let Q be a finite set of points in the plane. For any set P of points in the plane, SQ(P ) denotes the number of similar copies of Q contained in P . For a fixed n, Erdős and Purdy asked to determine the maximum possible value of SQ(P ), denoted by SQ(n), over all sets P of n points in the plane. We consider this problem when Q = △ is the set of vertices of an isosceles right triangle. We give exact solutions when n ≤ 9, and provide new upper and lower bounds for S△(n).
منابع مشابه
Online Circle and Sphere Packing
In this paper we consider the Online Bin Packing Problem in three variants: Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes. The two first ones receive an online sequence of circles (items) of different radii while the third one receive an online sequence of spheres (items) of different radii, and they want to pack the items into the minimum number of unit squares...
متن کاملSome 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...
متن کاملUnderstanding the Eigenstructure of Various Triangles
We examine the eigenstructure of generalized isosceles triangles and explore the possibilities of analytic solutions to the general eigenvalue problem in other triangles. Starting with work based off of Brian McCartin’s paper on equilateral triangles, we first explore the existence of analytic solutions within the space of all isosceles triangles. We find that this method only leads to consiste...
متن کاملEnclosing Isosceles Triangles and Related Problems ∗
Received (received date) Revised (revised date) Communicated by (Name) Given a set of n points in the plane, we show how to compute various enclosing isosceles triangles where different parameters such as area or perimeter are optimized. We then study a 3-dimensional version of the problem where we enclose a point set with a cone of fixed apex angle α.
متن کاملFinite configurations in sparse sets
Let E ⊆ Rn be a closed set of Hausdorff dimension α. For m ≥ n, let {B1, . . . , Bk} be n× (m−n) matrices. We prove that if the system of matrices Bj is non-degenerate in a suitable sense, α is sufficiently close to n, and if E supports a probability measure obeying appropriate dimensionality and Fourier decay conditions, then for a range of m depending on n and k, the set E contains a translat...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010