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On maximal families of subsets of a finite set

1981
*
Discrete Mathematics
*

Many

doi:10.1016/0012-365x(81)90059-5
fatcat:gpmjul6fczgyjmednhmgapvhmy
*of*tie well known conditions for*families**of**subsets**of**a*&n&e*set*can be described by certain expressions P. ... -tuple*of**subsets*X" Xz, . . , , X,*of**a*fi&e*set*R*of*carclinality consider 4 conditions which an ordered pair (X, Y) may or may not satisfy, namely r. ... Many*a*&ors have determined the*maximal*value*of*n if 9 satisfies some conditions P. Always ihey considered S : s an unordered n-tuple*of*distinct*subsets**of*R. ...##
###
Introduction to Matroids

2008
*
Formalized Mathematics
*

The paper includes elements

doi:10.2478/v10037-008-0040-0
fatcat:23abhwm2urhs5eg4gpsngmiwea
*of*the theory*of*matroids [23]. The formalization is done according to [12] . ... Let X be*a**finite**set*and let P be*a**subset**of*2 X .*One*can check that ProdMatroid P is*finite*. Let X be*a**set*. Observe that every partition*of*X is mutually-disjoint. ... The*family**of*M yielding*a**family**of**subsets**of*M is defined as follows:*One*can check that there exists*a**subset**family*structure which is strict, non empty, non void,*finite*, and*subset*-closed and has ...##
###
Applications of the notion of independence to problems of combinatorial analysis

1967
*
Journal of Combinatorial Theory
*

In the present paper we investigate properties

doi:10.1016/s0021-9800(67)80034-6
fatcat:az27zpdnurbphi7jgbrheym4ey
*of**a*general notion*of*independence and we use some*of*the results obtained to solve certain problems in combinatorial analysis concerned with the existence ...*of*systems*of*representatives. ... If at least*one**of*X, Y is*finite*, then*a**subset*X'*of*X is*maximal*with respect to the deltoid (X,*A*, Y) if and only if X' is linked to some*maximal**subset**of*Y. ...##
###
Subsets and Subgraphs with Maximal Properties

1959
*
Proceedings of the American Mathematical Society
*

Here tf>" is the

doi:10.2307/2033631
fatcat:kn34oxtzj5bafahzdjznvna5mq
*family**of*all*subsets**of*G containing at least*one*F as*a**subset*; for if*a**set*omits some elements in every F then its complement is*a*crossing*set*. ...*One*is often concerned with*subsets*H*of**a**set*G such that H is*maximal*among all those*subsets**of*G which have*a*certain property P. ...##
###
Subsets and subgraphs with maximal properties

1959
*
Proceedings of the American Mathematical Society
*

Here tf>" is the

doi:10.1090/s0002-9939-1959-0110643-7
fatcat:shhl7jzczjcttkw2w72niaceze
*family**of*all*subsets**of*G containing at least*one*F as*a**subset*; for if*a**set*omits some elements in every F then its complement is*a*crossing*set*. ...*One*is often concerned with*subsets*H*of**a**set*G such that H is*maximal*among all those*subsets**of*G which have*a*certain property P. ...##
###
Abstract Simplicial Complexes

2010
*
Formalized Mathematics
*

Simplicial Complexes In this article we define the notion

doi:10.2478/v10037-010-0013-y
fatcat:a4bcjaebijemflbmorswlvo564
*of*abstract simplicial complexes and operations*on*them. ... We introduce the following basic notions: simplex, face, vertex, degree, skeleton, subdivision and substructure, and prove some*of*their properties. ...*One*can verify that there exists*a**family**of**subsets**of*D which is*finite*,*subset*-closed, and*finite*-membered and has*a*non-empty element. Let us consider Y , X and let n be*a**finite**set*. ...##
###
Cocompactness in algebra and topology

1981
*
Journal of the Australian Mathematical Society
*

The first arises in the area

doi:10.1017/s1446788700024162
fatcat:ajksyepwszbe3mxhhykellukae
*of*topology, where J. de Groot and others' have studied spaces which are, in*a*certain sense, complementary to*a*given space. ... That is no*finite**subset**of*W intersects every*maximal*consistent*set*. ... This shows that U,"_i ^*A*i s*a**finite**subset**of**A*which intersects each*maximal*consistent*set*.*On*the other hand, suppose that (M x , T) is not compact. ...##
###
Maximal feebly compact spaces

1993
*
Topology and its Applications
*

*Maximal*feebly compact spaces (Le., feebly compact spaces possessing no strictly stronger feebly compact topology) are characterized, as are special classes (countably compact, semiregular, regular)

*of*...

*maximal*feebly compact spaces. ... Recall that if D is

*a*

*set*, an almost disjoint

*family*

*of*

*subsets*

*of*D is

*a*

*family*J

*of*infinite

*subsets*

*of*D such that distinct members

*of*d have

*finite*intersection. ...

##
###
Primitive shifts on ψ-spaces

2012
*
Topology and its Applications
*

*a*r t i c l e i n f o

*a*b s t r

*a*c t MSC: 54C35 47B38 ... If

*A*∈ F then treat

*A*also as

*a*

*subset*

*of*D.

*A*neighborhood base

*of*

*A*(treated as

*a*point

*of*X ) is

*a*collection

*of*

*sets*

*of*the form {

*A*} ∪ B where B ⊆

*A*and

*A*\ B is

*finite*. ... Let h : Q → B be

*one*-to-

*one*and onto. In Lemma 2.2 we constructed 2 c

*maximal*almost disjoint

*families*

*of*(infinite)

*subsets*

*of*Q . ...

##
###
MAD Families and the Ramsey Property

2021
*
Notices of the American Mathematical Society
*

Identifying

doi:10.1090/noti2314
fatcat:pkpmze3xljfjhiljgj7t4cajvu
*sets**of*natural numbers with real numbers is often*a*way to establish those connections. ... The study*of*collections*of**sets**of*natural numbers poses many interesting questions connected, sometimes surprisingly, to other areas*of*mathematics. ... Any infinite almost disjoint*family**of*infinite*subsets**of*ℕ is contained in*a**maximal*almost disjoint*family*. ...##
###
Appendix to "Extended Topology: Domain Finiteness"

1963
*
Indagationes Mathematicae (Proceedings)
*

Domain

doi:10.1016/s1385-7258(63)50037-7
fatcat:3344htpjdza3xmuvly6gss4osq
*Finite**Set*-valued Transformations Let .*A*be the class*of*all*subsets**of*space M and let .*A*o be the class*of*all*subsets**of**a*space Mo. Let t map .*A*into .Ao. ... The union*of*any*family**of*domain*finite*transformations is domain*finite*, and the intersection*of**a**finite**family*is domain*finite*. ... The union*of*any*family**of*domain*finite*transformations is domain*finite*, and the intersection*of**a**finite**family*is domain*finite*. ...##
###
What Is the Minimal Cardinal of a Family Which Shatters All d-Subsets of a Finite Set?
[chapter]

2016
*
Convexity and Discrete Geometry Including Graph Theory
*

Let S be

doi:10.1007/978-3-319-28186-5_27
fatcat:xqn3u3uesjaebkshttnmzoqccq
*a**finite**set**of*cardinal |S| = n and let 2 S denote its power*set*, i.e. the*set**of*its*subsets*.*A*d-*subset**of*S is*a**subset**of*S*of*cardinal d. Let F ⊆ 2 S and*A*⊆ S. ... The trace*of*F*on**A*is the*family*F*A*= {E ∩*A*; E ∈ F }.*One*says that F shatters*A*if F*A*= 2*A*. The VC-dimension*of*F is the*maximal*cardinal*of**a**subset**of*S that is shattered by F [7] . ...##
###
What is the minimal cardinal of a family which shatters all d-subsets of a finite set?
[article]

2015
*
arXiv
*
pre-print

What is the minimal cardinal

arXiv:1509.02087v1
fatcat:q5ynl2z52jftzcjdbx374olw3u
*of**a**family*which shatters all d-*subsets**of**a**finite**set*? ... Let S be*a**finite**set**of*cardinal |S| = n and let 2 S denote its power*set*, i.e. the*set**of*its*subsets*.*A*d-*subset**of*S is*a**subset**of*S*of*cardinal d. Let F ⊆ 2 S and*A*⊆ S. ... The trace*of*F*on**A*is the*family*F*A*= {E ∩*A*; E ∈ F }.*One*says that F shatters*A*if F*A*= 2*A*. The VC-dimension*of*F is the*maximal*cardinal*of**a**subset**of*S that is shattered by F [7] . ...##
###
Prime ideals in ultraproducts of commutative rings

2005
*
Journal of Algebra
*

We consider in particular prime ideals in ultraproducts

doi:10.1016/j.jalgebra.2004.11.004
fatcat:bkftb7gzcjexremyneqxlydiwy
*of*Noetherian rings, Krull domains,*finite*character rings and QR-domains. ... We describe classes*of*prime ideals in ultraproducts*of*commutative rings. ... Acknowledgment We thank the referee for*a*careful reading*of*the paper and many helpful comments. ...##
###
Corrigendum to "Some notes on weakly Whyburn spaces"

2004
*
Topology and its Applications
*

By the

doi:10.1016/j.topol.2003.11.003
fatcat:umyduw2bc5eflinkhg3ksodo2y
*maximality**of**A*the*set*E meets in fact uncountably many elements*of**A*in infinitely many points, since the trace*on*E*of*the*family**A*is*maximal*in E; but any uncountable*subset**of**A*has infinitely ... Let*A*be*a**maximal*almost disjoint*family*in ω, and let B be*a**maximal*almost disjoint*family**of*countable*subsets**of**A*. ... By the*maximality**of**A*the*set*E meets in fact uncountably many elements*of**A*in infinitely many points, since the trace*on*E*of*the*family**A*is*maximal*in E; but any uncountable*subset**of**A*has infinitely ...
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