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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 &~.  ... 
doi:10.1090/s0002-9939-1988-0955030-2 fatcat:jjksz25tm5drpifgwqhkgvl4pu

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 &~.  ... 
doi:10.2307/2047133 fatcat:wgwvcltvwbhalicw65vzqlu6m4

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  ... 
doi:10.2307/1995130 fatcat:fmtk7g5szbc33l5gxxxxuujibq

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  ... 
doi:10.1090/s0002-9947-1969-0249304-3 fatcat:app72cmrhzgpvm2a6e5t6njj54

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.  ... 
doi:10.1016/0022-247x(66)90106-5 fatcat:llqej3tm25huhfwzf5fbb6fmna

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.  ... 
doi:10.1090/s0002-9939-1989-0984798-5 fatcat:eubngckwuzggjowbfi4wafqhvq

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.  ... 
doi:10.2307/2047823 fatcat:5uupanjcnbdmtoeq57o7ivrrty

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 ?  ... 
doi:10.1016/0022-247x(70)90314-8 fatcat:n6po6u6nkvgwpgf4xhvmcisnze

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.  ... 
arXiv:1704.04343v2 fatcat:6dmn4n6qmbbtvlnhyb3vtwnvei

Systems of distant representatives in Euclidean space

Adrian Dumitrescu, Minghui Jiang
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.  ... 
doi:10.1016/j.jcta.2015.03.006 fatcat:qwphz5f7urf4bozc4qj4zcrrii

Systems of distant representatives in euclidean space

Adrian Dumitrescu, Minghui Jiang
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.  ... 
doi:10.1145/2493132.2462385 fatcat:7bhk2xc32jhxteifnuolwtmgta

Systems of distant representatives in euclidean space

Adrian Dumitrescu, Minghui Jiang
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.  ... 
doi:10.1145/2462356.2462385 dblp:conf/compgeom/DumitrescuJ13 fatcat:wpflned4fva5hglbnpf2rfqqei

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  ... 
doi:10.1016/0012-365x(92)90333-b fatcat:swvrkcvdyjfppetyf2uzek3bjq

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.  ... 
doi:10.1016/0012-365x(92)00567-b fatcat:ngni7knm2vc2zgpo72bfkfja44

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.  ... 
doi:10.1016/j.ejc.2006.07.001 fatcat:srnnmo2ej5b7tj3w7naxyo5fdy
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