Extremal Set Systems with Weakly Restricted Intersections

COMBINATORICA ◽  
1999 ◽  
Vol 19 (4) ◽  
pp. 567-587 ◽  
Author(s):  
Van H. Vu
Keyword(s):  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

scholarly journals Shattering-extremal set systems from Sperner families

2020 ◽  
Vol 276 ◽  
pp. 92-101
Author(s):  
Christopher Kusch ◽  
Tamás Mészáros
Keyword(s):  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

scholarly journals Shattering-Extremal Set Systems of VC Dimension at most 2

10.37236/4548 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai
Keyword(s):  
Vc Dimension ◽  
Set Systems ◽  
Dimension 2 ◽  
Extremal Set Systems ◽  
Extremal Set ◽  
Open Question

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 $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.

scholarly journals Extremal set systems with restricted k-wise intersections

2004 ◽  
Vol 105 (1) ◽  
pp. 143-159 ◽  
Author(s):  
Zoltán Füredi ◽  
Benny Sudakov
Keyword(s):  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

scholarly journals Shattering-Extremal Set Systems of Small VC-Dimension

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai
Keyword(s):  
Vc Dimension ◽  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

We say that a set system ℱ⊆2[n] shatters a given set S⊆[n] if 2S={F∩S:F∈ℱ}. The Sauer inequality states that in general, a set system ℱ shatters at least |ℱ| sets. Here, we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |ℱ| sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also, from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections.

scholarly journals VC Dimension and a Union Theorem for Set Systems

10.37236/8288 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Stijn Cambie ◽  
António Girão ◽  
Ross J. Kang
Keyword(s):  
Difference Operator ◽  
Negative Answer ◽  
Symmetric Difference ◽  
Vc Dimension ◽  
Compression Method ◽  
Set Systems ◽  
Theoretic Problem ◽  
Extremal Set ◽  
Positive Integers

Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most $(2^{d\bmod{k}}+o(1))\binom{n}{\lfloor d/k\rfloor}$. Here $\triangle$ denotes the symmetric difference operator. This is a $k$-fold generalisation of a result of Dvir and Moran, and it settles one of their questions.  A key insight is that, by a compression method, the problem is equivalent to an extremal set theoretic problem on $k$-wise intersection or union that was originally due to Erdős and Frankl. We also give an example of a family $\mathcal{A} \subset 2^{[n]}$ such that the VC dimension of $\mathcal{A}\cap \mathcal{A}$ and of $\mathcal{A}\cup \mathcal{A}$ are both at most $d$, while $\lvert \mathcal{A} \rvert = \Omega(n^d)$. This provides a negative answer to another question of Dvir and Moran.

scholarly journals Unavoidable subprojections in union-closed set systems of infinite breadth

2021 ◽  
Vol 94 ◽  
pp. 103311
Author(s):  
Yemon Choi ◽  
Mahya Ghandehari ◽  
Hung Le Pham
Keyword(s):  
Closed Set ◽  
Set Systems

scholarly journals The Fair Division of Hereditary Set Systems

2021 ◽  
Vol 9 (2) ◽  
pp. 1-19
Author(s):  
Z. Li ◽  
A. Vetta
Keyword(s):  
Recent Work ◽  
Polynomial Time ◽  
Simple Algorithm ◽  
Fair Division ◽  
Set Systems

We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form a hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3666 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.

scholarly journals The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

2021 ◽  
Author(s):  
Anne Driemel ◽  
André Nusser ◽  
Jeff M. Phillips ◽  
Ioannis Psarros
Keyword(s):  
Density Estimation ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Similarity Metrics ◽  
Vc Dimension ◽  
Set Systems ◽  
Polygonal Curves ◽  
The Individual ◽  
Curve Similarity ◽  
Metric Balls

AbstractThe Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

scholarly journals Non-Homogeneous Markov Set Systems

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 471
Author(s):  
P.-C.G. Vassiliou
Keyword(s):  
Asymptotic Behavior ◽  
Population Structure ◽  
Confidence Intervals ◽  
Limit Set ◽  
Stroke Patients ◽  
Relative Population ◽  
Set Systems ◽  
Population Structures ◽  
Basic Parameters ◽  
Initial Distributions

A more realistic way to describe a model is the use of intervals which contain the required values of the parameters. In practice we estimate the parameters from a set of data and it is natural that they will be in confidence intervals. In the present study, we study Non-Homogeneous Markov Systems (NHMS) processes for which the required basic parameters are in intervals. We call such processes Non-Homogeneous Markov Set Systems (NHMSS). First we study the set of the relative expected population structure of memberships and we prove that under certain conditions of convexity of the intervals of the parameters the set is compact and convex. Next, we establish that if the NHMSS starts with two different initial distributions sets and allocation probability sets under certain conditions, asymptotically the two expected relative population structures coincide geometrically fast. We continue proving a series of theorems on the asymptotic behavior of the expected relative population structure of a NHMSS and the properties of their limit set. Finally, we present an application for geriatric and stroke patients in a hospital and through it we solve problems that surface in an application.

Lower bound on the profile of degree pairs in cross-intersecting set systems

COMBINATORICA ◽  
2007 ◽  
Vol 27 (3) ◽  
pp. 399-405
Author(s):  
Zsuzsanna Szaniszló ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Set Systems
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