arXiv subjects

J. Pach

Publications and source records attributed to J. Pach.

3 recordsLinked to original sources

arXiv2021

Nearly equal distances in the plane, II

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1 \le \ell (1) \le \ell (2) < \ell (3) \le k$ that $|t_{\ell (3)} / (t_{\ell (1)} + t_{\ell (2)}) - 1| \ge \delta $. Then for $n \ge n_{k, \delta }$, the number of pairs $\{ p_i,p_j\} $, for which $d(p_i,p_j) \in [t_1, t_1 + 1] \cup \ldots \cup [t_k, t_k + 1] $, is at most $n^2/4 + C_{k,\delta }n$. This is sharp, up to the value of the constant $C_{k,\delta } > 0$.

math.CO↗

arXiv2019

Two nearly equal distances in $R^d$

A point set $P \subset {\Bbb{R}}^d$ is {\it separated} if the minimum distance between any two points in $P$ is at least $1$. For $d \ne 4,5,$ we determine, for every $t_1,t_2 \ge 1$, and for $n$ at least a suitable $n_d$, the maximum number of point pairs in a separated $n$-element point set in ${\Bbb{R}}^d$, with distances in the set $[t_1,t_1 + 1]\cup[t_2,t_2 + 1]$. For $d=4,5$ we establish a weaker, similar asymptotic estimate. Recently N. Frankl and A. Kupavskii have generalized this result to unions of $k\ge 2$ intervals. We also determine the maximum number of point pairs in an $n$-element point set in ${\Bbb{R}}^d$, whose distances belong to the union of $k \ge 2$ intervals of the form $[t_i, t_i(1 + \varepsilon)]$, where $t_i > 0$ and $\varepsilon > 0$ is small.

math.MG↗

arXiv2013

The Erdős-Hajnal conjecture for rainbow triangles

We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2 n) which uses at most two colors, and this bound is tight up to a constant factor. This verifies a conjecture of Hajnal which is a case of the multicolor generalization of the well-known Erdős-Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s at most r, we determine a constant c_{r,s} such that the following holds. Every r-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{r(r-1)/s(s-1)}(\log n)^{c_{r,s}}) which uses at most s colors, and this bound is tight apart from the implied constant factor. The proof of the lower bound utilizes Gallai's classification of rainbow-triangle free edge-colorings of the complete graph, a new weighted extension of Ramsey's theorem, and a discrepancy inequality in edge-weighted graphs. The proof of the upper bound uses Erdős' lower bound on Ramsey numbers by considering lexicographic products of 2-edge-colorings of complete graphs without large monochromatic cliques.

math.CO↗