ﻻ يوجد ملخص باللغة العربية
We consider the number of distinct distances between two finite sets of points in ${bf R}^k$, for any constant dimension $kge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ Omegaleft(minleft{ n^{2/3}m^{2/3},; frac{n^{10/11}m^{4/11}}{log^{2/11}m},; n^2,; m^2right}right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.
We prove the following generalised empty pentagon theorem: for every integer $ell geq 2$, every sufficiently large set of points in the plane contains $ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the
In this note we show that for each Latin square $L$ of order $ngeq 2$, there exists a Latin square $L eq L$ of order $n$ such that $L$ and $L$ differ in at most $8sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of
In this paper we investigate the extremal properties of the sum $$sum_{i=1}^n|MA_i|^{lambda},$$ where $A_i$ are vertices of a regular simplex, a cross-polytope (orthoplex) or a cube and $M$ varies on a sphere concentric to the sphere circumscribed ar
The emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-numb
We study ErdH oss distinct distances problem under $ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $Omega(n^{4/5})$ to $Omega(n^{6/7-epsilon})$, for any $epsilon>0$. We also characterize the sets that span an