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### On maximal families of subsets of a finite set

Hans-Dietrich O.H. Gronau
1981 Discrete Mathematics
Many 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

Grzegorz Bancerek, Yasunari Shidama
2008 Formalized Mathematics
The paper includes elements of the theory of matroids . The formalization is done according to  .  ...  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

L. Mirsky, Hazel Perfect
1967 Journal of Combinatorial Theory
In the present paper we investigate properties 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

Oystein Ore, T. S. Motzkin
1959 Proceedings of the American Mathematical Society
Here tf>" is the 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

Oystein Ore, T. S. Motzkin
1959 Proceedings of the American Mathematical Society
Here tf>" is the 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

Karol Pąk
2010 Formalized Mathematics
Simplicial Complexes In this article we define the notion 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

Graham J. Logan
1981 Journal of the Australian Mathematical Society
The first arises in the area 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

Jack R. Porter, Robert M. Stephenson, R. Grant Woods
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

A. Gutek, S.P. Moshokoa, M. Rajagopalan, K. Sundaresan
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

Carlos A. Di Prisco
2021 Notices of the American Mathematical Society
Identifying 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"

Preston C. Hammer
1963 Indagationes Mathematicae (Proceedings)
Domain 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]

Nicolas Chevallier, Augustin Fruchard
2016 Convexity and Discrete Geometry Including Graph Theory
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  .  ...

### What is the minimal cardinal of a family which shatters all d-subsets of a finite set? [article]

N. Chevallier
2015 arXiv   pre-print
What is the minimal cardinal 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  .  ...

### Prime ideals in ultraproducts of commutative rings

Bruce Olberding, Jay Shapiro
2005 Journal of Algebra
We consider in particular prime ideals in ultraproducts 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"

Franco Obersnel
2004 Topology and its Applications
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  ...  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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