相关论文: Shattering-extremal set systems of VC dimension at…

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S={F \cap S : F \in \mathcal{F}}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$…

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S= \{F~\cap~S:~F~\in~\mathcal{F}\}$. The Sauer-Shelah lemma states that in general, a set system $\mathcal{F}$ shatters at least…

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a set $S\subseteq [n]$ if every possible subset of $S$ appears as the intersection of $S$ with some element of $\mathcal{F}$ and we denote by $\text{Sh}(\mathcal{F})$ the…

We study the maximum size of a set system on $n$ elements whose trace on any $b$ elements has size at most $k$. We show that if for some $b \ge i \ge 0$ the shatter function $f_R$ of a set system $([n],R)$ satisfies $f_R(b) < 2^i(b-i+1)$…

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one…

We say that a set system $\mathcal{F}$ is $k$-completely hyperseparating if for any vertex $v$, there are at most $k$ sets in $\mathcal{F}$ with intersection $\{v\}$. We determine the minimum size of such set systems on an $n$-element…

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum…

Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the…

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$…

Many concepts from extremal set theory have analogues for families of permutations. This paper is concerned with the notion of shattering for permutations. A family $\mathcal{P}$ of permutations of an $n$-element set $X$ shatters a $k$-set…

An ordered variant of the well-known set theory concept of shattering was introduced by Anstee, R\'onyai, and Sali. In this paper, we prove several new results related to order shattering. Given a family $\mathcal F$ of subsets of $[n]$, we…

The well-known Sauer lemma states that a family $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension at most $d$ has size at most $\sum_{i=0}^d\binom{n}{i}$. We obtain both random and explicit constructions to prove that the corresponding…

Let $\mbox{ V}(n,d,q)$ stand for the $q$--ary Hamming spheres. Let $\mbox{ C}\subseteq (q)^n$ denote a tuple system such that $\mbox{ C}\subseteq \cup_{i=0}^s \mbox{ V}(n,d_i,q)$, where $d_1<\ldots <d_s$. We give here a general upper bound…

A set system $\mathcal{F}$ is $t$-\textit{intersecting}, if the size of the intersection of every pair of its elements has size at least $t$. A set system $\mathcal{F}$ is $k$-\textit{Sperner}, if it does not contain a chain of length…

We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied…

2+1D multi-component $U(1)$ gauge theories with a Chern-Simons (CS) term provide a simple and complete characterization of 2+1D Abelian topological orders. In this paper, we extend the theory by taking the number of component gauge fields…

A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most…

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained…

In computer science, combinatorics, and model theory, the VC dimension is a central notion underlying far-reaching topics such as error rate for decision rules, combinatorial measurements of classes of finite structures, and neo-stability…

We have found the most general extension of the celebrated Sauer, Perles and Shelah, Vapnik and Chervonenkis result from 0-1 sequences to $k$-ary codes still giving a polynomial bound. Let $\mathcal{C}\subseteq \{0,1,..., k-1}^n$ be a…