Related papers: Intersections of shifted sets

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random…

We define and consider k-distant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng…

Though a convergence space is connected if and only if its topological modification is connected, connected subsets differ for the convergence and for its topological modification. We explore for what subsets connectedness for the…

Let $G_{k,n}$ be a group of permutations of $kn$ objects which permutes things independently in disjoint blocks of size $k$ and then permutes the blocks. We investigate the probabilistic and/or enumerative aspects of random elements of…

We prove the complete intersection theorem and complete nontrivial-intersection theorem for systems of set partitions

Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…

We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the…

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number.

We use the fact that certain cosets of the stabilizer of points are pairwise conjugate in a symmetric group $S_n$ in order to construct recurrence relations for enumerating certain subsets of $S_n$. Occasionally one can find `closed form'…

The random reversal graph offers new perspectives, allowing to study the connectivity of genomes as well as their most likely distance as a function of the reversal rate. Our main result shows that the structure of the random reversal graph…

We show that intersection graphs of compact convex sets in R^n of bounded aspect ratio have asymptotic dimension at most 2n+1. More generally, we show this is the case for intersection graphs of systems of subsets of any metric space of…

We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous…

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szab\'{o} Theorem…

We study the intersecting family process initially studied in \cite{BCFMR}. Here $k=k(n)$ and $E_1,E_2,\ldots,E_m$ is a random sequence of $k$-sets from $\binom{[n]}{k}$ where $E_{r+1}$ is uniformly chosen from those $k$-sets that are not…

Menger's theorem says that, for $k\ge0$, if $S, T$ are sets of vertices in a graph $G$, then either there are $k + 1$ vertex-disjoint paths between $S$ and $T$, or there is a set X of at most $k$ vertices such that every $S$-$T$ path passes…

If we are given a connected finite graph $G$ and a subset of its vertices $V_{0}$, we define a distance-residual graph as a graph induced on the set of vertices that have the maximal distance from $V_{0}$. Some properties and examples of…

Given a set $S$ of $n$ points in $\mathbb{R}^d$, a $k$-set is a subset of $k$ points of $S$ that can be strictly separated by a hyperplane from the remaining $n-k$ points. Similarly, one may consider $k$-facets, which are hyperplanes that…

We show the linear convergence of Dykstra's algorithm for sets intersecting in a manner slightly stronger than the usual constraint qualifications.

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various…

Let $V \subset \mathbb{R}$ be a finite set with $|V| = n $ and suppose we are given each pairwise distance independently with probability $p$. We show that if $p = (1+\epsilon)/n$, for some fixed $\epsilon >0$, then we can reconstruct a…