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### Simultaneous systems of representatives for families of finite sets

Melvyn B. Nathanson
1988 Proceedings of the American Mathematical Society
The set A is a system of representatives for S? if X f~l S% ^ 0 for all i G I. If A is a system of representatives for 5?  ...  but no proper subset of A is a system of representatives for S", then A is a minimal system of representatives for S?. Let M'S?) denote the number of minimal systems of representatives for S*.  ...  Let N(S',j7') denote the number of sets A such that A is a minimal system of representatives for c5^ and X is simultaneously a system of representatives for &~.  ...

### Simultaneous Systems of Representatives for Families of Finite Sets

Melvyn B. Nathanson
1988 Proceedings of the American Mathematical Society
The set A is a system of representatives for S? if X f~l S% ^ 0 for all i G I. If A is a system of representatives for 5?  ...  but no proper subset of A is a system of representatives for S", then A is a minimal system of representatives for S?. Let M'S?) denote the number of minimal systems of representatives for S*.  ...  Let N(S',j7') denote the number of sets A such that A is a minimal system of representatives for c5^ and X is simultaneously a system of representatives for &~.  ...

### A Very General Theorem on Systems of Distinct Representatives

Richard A. Brualdi
1969 Transactions of the American Mathematical Society
Hall proved his now celebrated theorem for the existence of a system of distinct representatives of a finite family of sets. In a no less significant paper M. Hall, Jr. (in 1948) extended P.  ...  The theorem we prove contains as special cases (that is, without further refinement) all theorems that we know which assert the existence of a system of distinct representatives of a given family of sets  ...  For D a locally finite directed graph, if ap = a'p and bp = b'p for all pe P, then Theorem 8 describes when D has a subgraph for which the indegree and outdegree of each point p is ap and bp respectively  ...

### A very general theorem on systems of distinct representatives

Richard A. Brualdi
1969 Transactions of the American Mathematical Society
Hall proved his now celebrated theorem for the existence of a system of distinct representatives of a finite family of sets. In a no less significant paper M. Hall, Jr. (in 1948) extended P.  ...  The theorem we prove contains as special cases (that is, without further refinement) all theorems that we know which assert the existence of a system of distinct representatives of a given family of sets  ...  For D a locally finite directed graph, if ap = a'p and bp = b'p for all pe P, then Theorem 8 describes when D has a subgraph for which the indegree and outdegree of each point p is ap and bp respectively  ...

### Systems of representatives

L Mirsky, Hazel Perfect
1966 Journal of Mathematical Analysis and Applications
We shall then call (xi : i E I) a system of distinct representatives (or a transversal) of the given family of sets.  ...  Now a family need not, of course, possess a system of distinct representatives, and an interesting question is to seek conditions for the existence of such a system.  ...  Symmetrization of Hall's Problem Suppose that the family (ri : 1 < i < n) of finite sets possesses a system of distinct representatives.  ...

### Representatives for finite sets

Xing De Jia
1989 Proceedings of the American Mathematical Society
Nathanson [ 1 ], concerning simultaneous systems of representatives for two families of finite sets.  ...  -{S¡} be a family of nonempty sets. The set X is a system of representatives for 5? if X n 5J / 0 for every S¡ in S?. If X is a system of representatives for 5?  ...  5 = {Sj} is a family of í nonempty, distinct sets S{ with \S¡\ < h for all i ; (ii) y = {Tj} is a family of t nonempty, pairwise disjoint sets Tj with \Tj\ <k for all ;'; (iii) S¡ is not a subset of T.  ...

### Representatives for Finite Sets

Xing-De Jia
1989 Proceedings of the American Mathematical Society
Nathanson [ 1 ], concerning simultaneous systems of representatives for two families of finite sets.  ...  -{S¡} be a family of nonempty sets. The set X is a system of representatives for 5? if X n 5J / 0 for every S¡ in S?. If X is a system of representatives for 5?  ...  5 = {Sj} is a family of í nonempty, distinct sets S{ with \S¡\ < h for all i ; (ii) y = {Tj} is a family of t nonempty, pairwise disjoint sets Tj with \Tj\ <k for all ;'; (iii) S¡ is not a subset of T.  ...

### Systems of distinct representatives, II

Phillip A Ostrand
1970 Journal of Mathematical Analysis and Applications
A system of distinct representatives (abbreviated SDR) of F is an indexed set {xi 1 i ~1) of distinct elements for which xi E Fi for each i E I.  ...  INTRODUCTION Given a finite family of finite sets which has a system of distinct representatives, how many different systems does it have ?  ...

### The de Bruijn-Erdös theorem in incidence geometry via Ph. Hall's marriage theorem [article]

Nikolai V. Ivanov
2017 arXiv   pre-print
The paper is devoted to a proof of the de Bruijn-Erd\"os theorem in incidence geometry based on the Ph. Hall's marriage theorem (the theorem about the systems of distinct representatives).  ...  Systems of distinct representatives. A (finite) family of subsets of a set S is a map ϕ : I −→ 2 S from a finite set I to the power set 2 S . Usually ϕ(i ) is denoted by ϕ i .  ...  The theorem applies to ψ and implies that there exists a system of distinct representatives g : K −→ S for ψ.  ...

### Systems of distant representatives in Euclidean space

2015 Journal of combinatorial theory. Series A
Given a finite family of sets, Hall's classical marriage theorem provides a necessary and sufficient condition for the existence of a system of distinct representatives for the sets in the family.  ...  Here we extend this result to a geometric setting: given a finite family of objects in the Euclidean space (e.g., convex bodies), we provide a sufficient condition for the existence of a system of distinct  ...  A system of distinct representatives (SDR) for A is an indexed set S = {a j | j ∈ J} of distinct elements with a j ∈ A j , ∀j ∈ J.  ...

### Systems of distant representatives in euclidean space

2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
Given a finite family of sets, Hall's classical marriage theorem provides a necessary and sufficient condition for the existence of a system of distinct representatives for the sets in the family.  ...  Here we extend this result to a geometric setting: given a finite family of objects in the Euclidean space (e.g., convex bodies), we provide a sufficient condition for the existence of a system of distinct  ...  A system of distinct representatives (SDR) for A is an indexed set S = {a j | j ∈ J} of distinct elements with a j ∈ A j , ∀j ∈ J.  ...

### Systems of distant representatives in euclidean space

2013 Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
Given a finite family of sets, Hall's classical marriage theorem provides a necessary and sufficient condition for the existence of a system of distinct representatives for the sets in the family.  ...  Here we extend this result to a geometric setting: given a finite family of objects in the Euclidean space (e.g., convex bodies), we provide a sufficient condition for the existence of a system of distinct  ...  A system of distinct representatives (SDR) for A is an indexed set S = {a j | j ∈ J} of distinct elements with a j ∈ A j , ∀j ∈ J.  ...

### The notion and basic properties of M-transversals

Martin Kochol
1992 Discrete Mathematics
., The notion and basic properties of M-transversals, Discrete Mathematics 104 (1992) 191-196. 0012-365X/92/\$05.00 0 1992-Elsevier Science Publishers B.V. All rights reserved  ...  If, in addition, xi #xi, for any i #j, then (xi: i E I) is called the system of distinct representatives (SDR) of &.  ...  of Sp; then the set X = {x,: i E I} (i.e., the set of distinct elements of the system (x,: i E I)) is called the M-transversal of _cI, (If U, is the k-uniform matroid of rank k, then the Uk-transversal  ...

### Compatible systems of representatives

Martin Kochol
1994 Discrete Mathematics
We give a necessary and sufficient condition for G to have k pairwise compatible systems of representatives with at least d edges.  ...  Let G be a bipartite graph and assume that for any vertex v of G a strongly base orderable matroid is given on the set of edges adjacent with v.  ...  systems of distinct representatives.  ...

### Minimal paths and cycles in set systems

Dhruv Mubayi, Jacques Verstraëte
2007 European journal of combinatorics (Print)
Let f r (n, k) be the maximum size of a family of r -sets of an n element set containing no minimal k-cycle. Our results imply that for fixed r, k ≥ 3, where = (k − 1)/2 .  ...  A minimal k-cycle is a family of sets A 0 , . . . , A k−1 for which A i ∩ A j = ∅ if and only if i = j or i and j are consecutive modulo k.  ...  Acknowledgments The authors thank the referee for helpful comments which included shortening the proof of Theorem 1.3 Case 1.  ...
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