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Simultaneous systems of representatives for families of finite sets

1988
*
Proceedings of the American Mathematical Society
*

The

doi:10.1090/s0002-9939-1988-0955030-2
fatcat:jjksz25tm5drpifgwqhkgvl4pu
*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

1988
*
Proceedings of the American Mathematical Society
*

The

doi:10.2307/2047133
fatcat:wgwvcltvwbhalicw65vzqlu6m4
*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

1969
*
Transactions of the American Mathematical Society
*

Hall proved his now celebrated theorem

doi:10.2307/1995130
fatcat:fmtk7g5szbc33l5gxxxxuujibq
*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 ...##
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A very general theorem on systems of distinct representatives

1969
*
Transactions of the American Mathematical Society
*

Hall proved his now celebrated theorem

doi:10.1090/s0002-9947-1969-0249304-3
fatcat:app72cmrhzgpvm2a6e5t6njj54
*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 ...##
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Systems of representatives

1966
*
Journal of Mathematical Analysis and Applications
*

We shall then call (xi : i E I) a

doi:10.1016/0022-247x(66)90106-5
fatcat:llqej3tm25huhfwzf5fbb6fmna
*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*. ...##
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Representatives for finite sets

1989
*
Proceedings of the American Mathematical Society
*

Nathanson [ 1 ], concerning simultaneous

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

1989
*
Proceedings of the American Mathematical Society
*

Nathanson [ 1 ], concerning simultaneous

doi:10.2307/2047823
fatcat:5uupanjcnbdmtoeq57o7ivrrty
*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

1970
*
Journal of Mathematical Analysis and Applications
*

A

doi:10.1016/0022-247x(70)90314-8
fatcat:n6po6u6nkvgwpgf4xhvmcisnze
*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 ? ...##
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The de Bruijn-Erdös theorem in incidence geometry via Ph. Hall's marriage theorem
[article]

2017
*
arXiv
*
pre-print

The paper is devoted to a proof

arXiv:1704.04343v2
fatcat:6dmn4n6qmbbtvlnhyb3vtwnvei
*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. ...##
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Systems of distant representatives in Euclidean space

2015
*
Journal of combinatorial theory. Series A
*

Given a finite

doi:10.1016/j.jcta.2015.03.006
fatcat:qwphz5f7urf4bozc4qj4zcrrii
*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. ...##
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Systems of distant representatives in euclidean space

2013
*
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
*

Given a finite

doi:10.1145/2493132.2462385
fatcat:7bhk2xc32jhxteifnuolwtmgta
*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. ...##
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Systems of distant representatives in euclidean space

2013
*
Proceedings of the 29th annual symposium on Symposuim on computational geometry - SoCG '13
*

Given a finite

doi:10.1145/2462356.2462385
dblp:conf/compgeom/DumitrescuJ13
fatcat:wpflned4fva5hglbnpf2rfqqei
*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. ...##
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The notion and basic properties of M-transversals

1992
*
Discrete Mathematics
*

., The notion and basic properties

doi:10.1016/0012-365x(92)90333-b
fatcat:swvrkcvdyjfppetyf2uzek3bjq
*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 ...##
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Compatible systems of representatives

1994
*
Discrete Mathematics
*

We give a necessary and sufficient condition

doi:10.1016/0012-365x(92)00567-b
fatcat:ngni7knm2vc2zgpo72bfkfja44
*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*. ...##
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Minimal paths and cycles in set systems

2007
*
European journal of combinatorics (Print)
*

Let f r (n,

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