Related papers: Unit Distance Problems

We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is…

Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…

Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must…

Let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic curves in $\mathbb{R}^d$. Assume that $\gamma_i$ is neither contained in a hyperplane nor in a quadric surface in $\mathbb{R}^d$, for each $i=1,2$. We show that for…

Applications in machine learning and data mining require computing pairwise Lp distances in a data matrix A. For massive high-dimensional data, computing all pairwise distances of A can be infeasible. In fact, even storing A or all pairwise…

We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i \ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of Fukuda…

We consider a problem posed by Erd\H{o}s, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the…

We investigate a restriction of Paul Erdos' well-known problem from 1936 on the discrepancy of homogeneous arithmetic progressions. We restrict our attention to a finite set S of homogeneous arithmetic progressions, and ask when the…

We consider the problem of computing the (two-sided) Hausdorff distance between the unit $\ell_{p_{1}}$ and $\ell_{p_{2}}$ norm balls in finite dimensional Euclidean space for $1 \leq p_1 < p_2 \leq \infty$, and derive a closed-form formula…

The maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=6$, and constant dimension $k=4$ is $257$, where the $2$ isomorphism types are extended lifted maximum rank distance codes. In…

In this paper, we prove that a set of $N$ points in ${\bf R}^2$ has at least $c{N \over \log N}$ distinct distances, thus obtaining the sharp exponent in a problem of Erd\"os. We follow the set-up of Elekes and Sharir which, in the spirit…

We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower…

We consider a finite fields version of the Erd\H{o}s-Falconer distance problem for two different sets. In a certain range for the sizes of the two sets we obtain results of the conjectured order of magnitude.

We show that the number of unit distances determined by n points in R^3 is O(n^{3/2}), slightly improving the bound of Clarkson et al. established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth…

A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\H{o}s, Ko, and Rado determines the maximum size of an intersecting family of…

Based on combinatorics, we evaluate the upper bounds for the number of solutions to spatially coupled Sudokus, which are popular logic puzzles.

The maximum cut problem for a quintic del Pezzo surface ${\rm Bl}_{4}(\mathbb{P}^2)$ asks: Among all partitions of the 10 exceptional curves into two disjoint sets, what is the largest possible number of pairwise intersections? In this…

Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\leq i\leq p$) is at…

Let R be a family of n axis-parallel rectangles with packing number p-1, meaning that among any p of the rectangles, there are two with a non-empty intersection. We show that the union complexity of R is at most O(n+p^2), and that the…

Erd\H{o}s asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime…