Hall's Marriage Theorem for correspondence
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Let $A={A_1,....A_n}$ be a collection of subsets of a finite set $X$. A selection for $A$ is the image of an injective function $f:Ato X$ such that $ f(A_i)in A_i$ for every $A_iin A$.
Hall's marriage theorem shows that , $A$ has a selection if and only if for each subset $Ssubseteq A$,
$$|S|leq |cup_{iin S} A_i|.$$
I wonder if it is possible to generalize this result and obtain a similar condition for a choice correspondence $f:Arightrightarrows X $ that selects for each $A_i$ more than one elements, i.e. $f(A_i)subset A_i$ and $|f(A_i)|=2$ and all selected elements are distinct.
Any ideas or suggestions?
combinatorics discrete-mathematics graph-theory
asked 21 hours ago
sam
132
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up vote
2
down vote
favorite
Let $A={A_1,....A_n}$ be a collection of subsets of a finite set $X$. A selection for $A$ is the image of an injective function $f:Ato X$ such that $ f(A_i)in A_i$ for every $A_iin A$.
Hall's marriage theorem shows that , $A$ has a selection if and only if for each subset $Ssubseteq A$,
$$|S|leq |cup_{iin S} A_i|.$$
I wonder if it is possible to generalize this result and obtain a similar condition for a choice correspondence $f:Arightrightarrows X $ that selects for each $A_i$ more than one elements, i.e. $f(A_i)subset A_i$ and $|f(A_i)|=2$ and all selected elements are distinct.
Any ideas or suggestions?
combinatorics discrete-mathematics graph-theory
asked 21 hours ago
sam
132
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $A={A_1,....A_n}$ be a collection of subsets of a finite set $X$. A selection for $A$ is the image of an injective function $f:Ato X$ such that $ f(A_i)in A_i$ for every $A_iin A$.
Hall's marriage theorem shows that , $A$ has a selection if and only if for each subset $Ssubseteq A$,
$$|S|leq |cup_{iin S} A_i|.$$
I wonder if it is possible to generalize this result and obtain a similar condition for a choice correspondence $f:Arightrightarrows X $ that selects for each $A_i$ more than one elements, i.e. $f(A_i)subset A_i$ and $|f(A_i)|=2$ and all selected elements are distinct.
Any ideas or suggestions?
combinatorics discrete-mathematics graph-theory
asked 21 hours ago
sam
132
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $A={A_1,....A_n}$ be a collection of subsets of a finite set $X$. A selection for $A$ is the image of an injective function $f:Ato X$ such that $ f(A_i)in A_i$ for every $A_iin A$.
Hall's marriage theorem shows that , $A$ has a selection if and only if for each subset $Ssubseteq A$,
$$|S|leq |cup_{iin S} A_i|.$$
I wonder if it is possible to generalize this result and obtain a similar condition for a choice correspondence $f:Arightrightarrows X $ that selects for each $A_i$ more than one elements, i.e. $f(A_i)subset A_i$ and $|f(A_i)|=2$ and all selected elements are distinct.
Any ideas or suggestions?
combinatorics discrete-mathematics graph-theory
combinatorics discrete-mathematics graph-theory
asked 21 hours ago
sam
132
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 21 hours ago
sam
132
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 21 hours ago
sam
132
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 21 hours ago
sam
132
asked 21 hours ago
sam
132
132
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
sam is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Yes, this is a straightforward extension. Instead of the sequence
$$A_1,A_2,ldots,A_n$$
consider the sequence
$$A_1,A_1,A_2,A_2,ldots,A_n,A_n.$$
The Hall condition for this sequence amounts to
$$left|bigcup_{iin S}A_iright|ge2|S|.$$
answered 20 hours ago
Lord Shark the Unknown
98k958130
Great. Thanks for your answer!
– sam
20 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Yes, this is a straightforward extension. Instead of the sequence
$$A_1,A_2,ldots,A_n$$
consider the sequence
$$A_1,A_1,A_2,A_2,ldots,A_n,A_n.$$
The Hall condition for this sequence amounts to
$$left|bigcup_{iin S}A_iright|ge2|S|.$$
answered 20 hours ago
Lord Shark the Unknown
98k958130
Great. Thanks for your answer!
– sam
20 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
add a comment |
up vote
2
down vote
accepted
Yes, this is a straightforward extension. Instead of the sequence
$$A_1,A_2,ldots,A_n$$
consider the sequence
$$A_1,A_1,A_2,A_2,ldots,A_n,A_n.$$
The Hall condition for this sequence amounts to
$$left|bigcup_{iin S}A_iright|ge2|S|.$$
answered 20 hours ago
Lord Shark the Unknown
98k958130
Great. Thanks for your answer!
– sam
20 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Yes, this is a straightforward extension. Instead of the sequence
$$A_1,A_2,ldots,A_n$$
consider the sequence
$$A_1,A_1,A_2,A_2,ldots,A_n,A_n.$$
The Hall condition for this sequence amounts to
$$left|bigcup_{iin S}A_iright|ge2|S|.$$
answered 20 hours ago
Lord Shark the Unknown
98k958130
Yes, this is a straightforward extension. Instead of the sequence
$$A_1,A_2,ldots,A_n$$
consider the sequence
$$A_1,A_1,A_2,A_2,ldots,A_n,A_n.$$
The Hall condition for this sequence amounts to
$$left|bigcup_{iin S}A_iright|ge2|S|.$$
answered 20 hours ago
Lord Shark the Unknown
98k958130
answered 20 hours ago
Lord Shark the Unknown
98k958130
answered 20 hours ago
Lord Shark the Unknown
98k958130
answered 20 hours ago
Lord Shark the Unknown
98k958130
98k958130
Great. Thanks for your answer!
– sam
20 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
add a comment |
Great. Thanks for your answer!
– sam
20 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
Great. Thanks for your answer!
– sam
20 hours ago
Great. Thanks for your answer!
– sam
20 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
@sam This doesn't work very well with your (incorrect) definition of a choice function. You should define a choice function as an injection $f:{1,2,dots,n}to X$ instead of an injection $f:{A_1,A_2,dots,A_n}to X$ for just this reason, that you don't want to require the sets $A_i$ to be distinct.
– bof
13 hours ago
add a comment |
sam is a new contributor. Be nice, and check out our Code of Conduct.
sam is a new contributor. Be nice, and check out our Code of Conduct.
sam is a new contributor. Be nice, and check out our Code of Conduct.
sam is a new contributor. Be nice, and check out our Code of Conduct.
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