Related papers: Reasonable Bounds for Combinatorial Lines of Lengt…

Let $\Gamma$ be an abelian group and $g \geq h \geq 2$ be integers. A set $A \subset \Gamma$ is a $C_h[g]$-set if given any set $X \subset \Gamma$ with $|X| = k$, and any set $\{ k_1 , \dots , k_g \} \subset \Gamma$, at least one of the…

In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple…

We show that the polyhedron defined as the convex hull of the lattice points above the hyperbola $\left\{xy = n\right\}$ has between $\Omega(n^{1/3})$ and $O(n^{1/3} \log n)$ vertices. The same bounds apply to any hyperbola with rational…

We combinatorially prove that the number $R(n,k)$ of permutations of length $n$ having $k$ runs is a log-concave sequence in $k$, for all $n$. We also give a new combinatorial proof for the log-concavity of the Eulerian numbers.

We present a near-linear time approximation algorithm for the subtrajectory cluster problem of $c$-packed trajectories. The problem involves finding $m$ subtrajectories within a given trajectory $T$ such that their Fr\'echet distances are…

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We address the question of determining the maximum size of equiangular line sets in $\mathbb{R}^n$, using semidefinite programming…

It is established that for any finite set of positive real numbers $A$, we have $$|A/A+A| \gg \frac{|A|^{\frac{3}{2}+\frac{1}{26}}}{\log^{1/2}|A|}.$$

We show that the number of geometric permutations of an arbitrary collection of $n$ pairwise disjoint convex sets in $\mathbb{R}^d$, for $d\geq 3$, is $O(n^{2d-3}\log n)$, improving Wenger's 20 years old bound of $O(n^{2d-2})$.

Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$,…

We investigate decomposable combinatorial labeled structures more fully, focusing on the exp-log class of type a=1 or 1/2. For instance, the modal length of the second longest cycle in a random n-permutation is (0.2350...)n, whereas the…

Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we…

Let $L$ be a set of $n$ lines in $\reals^d$, for $d\ge 3$. A {\em joint} of $L$ is a point incident to at least $d$ lines of $L$, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum…

We survey some of the mechanisms used to prove that naturally defined sequences in combinatorics are log-concave. Among these mechanisms are Alexandrov's inequality for mixed discriminants, the Alexandrov Fenchel inequality for mixed…

Let T_n denote the set of log canonical thresholds of pairs (X,Y), with X a nonsingular variety of dimension n, and Y a nonempty closed subscheme of X. Using non-standard methods, we show that every limit of a decreasing sequence in T_n…

Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

Let $\Gamma$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(\Gamma)$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n} =…

A matching is compatible to two or more labeled point sets of size $n$ with labels $\{1,\dots,n\}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to…

Let $I(n,l)$ denote the maximum possible number of incidences between $n$ points and $l$ lines. It is well known that $I(n,l) = \Theta(n^{2/3}l^{2/3} + n + l)$. Let $c_{\mathrm{SzTr}}$ denote the lower bound on the constant of…

We prove that a general complete intersection of dimension $n$, codimension $c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if $c \geq 2n-2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in $N$ and…