Prove every infinite directed graph has infinite strict linear order or…
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I'm trying to prove the following: Every infinite directed graph has
an infinite subset of vertices that induces one of the following:
- a strict linear order
- a weak linear order
- an independent set without loops
- loops at each vertex and otherwise no edges
I was thinking of seeing $(V,E)$ as a set with a relation $(P,leq)$ and since it can be seen as an infinite poset with reachability I can use Ramsey's theorem by coloring every pair as either connected or not, then we have an infinite chain or infinite antichain.
The chain is an infinite path, if it has loops at every vertex then it is reflexive and a weak linear order. If not then it is a strong linear order.
If it has an infinite anti chain, it is possible that the vertices in the antichain have loops aka $v leq v, forall vin A$, $A$ antichain, if not then they are completely independent with $vnot in Xin {Achoose 2 }$, for all edges $X$.
I'm not sure if these thoughts can be put in a coherent proof.
proof-verification graph-theory ramsey-theory
asked Jan 9 at 11:28
orangeorange
729316
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add a comment |
$begingroup$
I'm trying to prove the following: Every infinite directed graph has
an infinite subset of vertices that induces one of the following:
- a strict linear order
- a weak linear order
- an independent set without loops
- loops at each vertex and otherwise no edges
I was thinking of seeing $(V,E)$ as a set with a relation $(P,leq)$ and since it can be seen as an infinite poset with reachability I can use Ramsey's theorem by coloring every pair as either connected or not, then we have an infinite chain or infinite antichain.
The chain is an infinite path, if it has loops at every vertex then it is reflexive and a weak linear order. If not then it is a strong linear order.
If it has an infinite anti chain, it is possible that the vertices in the antichain have loops aka $v leq v, forall vin A$, $A$ antichain, if not then they are completely independent with $vnot in Xin {Achoose 2 }$, for all edges $X$.
I'm not sure if these thoughts can be put in a coherent proof.
proof-verification graph-theory ramsey-theory
asked Jan 9 at 11:28
orangeorange
729316
$endgroup$
$begingroup$
If every pair of vertices is joined by directed edges in both directions, thhen none of your four alternatives holds. I guess you are assuming some antisymmetry: " if edges $ab$ and $ba$ are both present, then $a=b$" or something like that. I don't think that is the usual meaning of the term "directed graph".
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– bof
Jan 9 at 12:30
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Would such a graph contain an infinite vertex set that would induce an infinite clique?
$endgroup$
– orange
Jan 9 at 12:35
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Here is how to use Ramsey's theorem. Choose an infinite sequence $v_1,v_2,dots$ of distinct vertices. Color the unordered pairs of natural numbers with $8$ colors as follows. Suppose $i,jinmathbb N$, $ilt j$. Give the pair ${i,j}$ the color $(x,y,z)$ where $x$ is the truth value of "there is a loop at $v_i$", $y$ is the truth value of "there is a directed edge $v_iv_j$", and $z$ is the truth value of "there is a directed edge $v_jv_i$". Apply Ramsey's theorem to get an infinite monochromatic set. Which of the $8$ colors are possible depends on what you assume about the starting graph.
$endgroup$
– bof
Jan 9 at 12:37
add a comment |
$begingroup$
I'm trying to prove the following: Every infinite directed graph has
an infinite subset of vertices that induces one of the following:
- a strict linear order
- a weak linear order
- an independent set without loops
- loops at each vertex and otherwise no edges
I was thinking of seeing $(V,E)$ as a set with a relation $(P,leq)$ and since it can be seen as an infinite poset with reachability I can use Ramsey's theorem by coloring every pair as either connected or not, then we have an infinite chain or infinite antichain.
The chain is an infinite path, if it has loops at every vertex then it is reflexive and a weak linear order. If not then it is a strong linear order.
If it has an infinite anti chain, it is possible that the vertices in the antichain have loops aka $v leq v, forall vin A$, $A$ antichain, if not then they are completely independent with $vnot in Xin {Achoose 2 }$, for all edges $X$.
I'm not sure if these thoughts can be put in a coherent proof.
proof-verification graph-theory ramsey-theory
asked Jan 9 at 11:28
orangeorange
729316
$endgroup$
I'm trying to prove the following: Every infinite directed graph has
an infinite subset of vertices that induces one of the following:
- a strict linear order
- a weak linear order
- an independent set without loops
- loops at each vertex and otherwise no edges
I was thinking of seeing $(V,E)$ as a set with a relation $(P,leq)$ and since it can be seen as an infinite poset with reachability I can use Ramsey's theorem by coloring every pair as either connected or not, then we have an infinite chain or infinite antichain.
The chain is an infinite path, if it has loops at every vertex then it is reflexive and a weak linear order. If not then it is a strong linear order.
If it has an infinite anti chain, it is possible that the vertices in the antichain have loops aka $v leq v, forall vin A$, $A$ antichain, if not then they are completely independent with $vnot in Xin {Achoose 2 }$, for all edges $X$.
I'm not sure if these thoughts can be put in a coherent proof.
proof-verification graph-theory ramsey-theory
proof-verification graph-theory ramsey-theory
asked Jan 9 at 11:28
orangeorange
729316
asked Jan 9 at 11:28
orangeorange
729316
asked Jan 9 at 11:28
orangeorange
729316
asked Jan 9 at 11:28
orangeorange
729316
asked Jan 9 at 11:28
orangeorange
729316
729316
$begingroup$
If every pair of vertices is joined by directed edges in both directions, thhen none of your four alternatives holds. I guess you are assuming some antisymmetry: " if edges $ab$ and $ba$ are both present, then $a=b$" or something like that. I don't think that is the usual meaning of the term "directed graph".
$endgroup$
– bof
Jan 9 at 12:30
$begingroup$
Would such a graph contain an infinite vertex set that would induce an infinite clique?
$endgroup$
– orange
Jan 9 at 12:35
$begingroup$
Here is how to use Ramsey's theorem. Choose an infinite sequence $v_1,v_2,dots$ of distinct vertices. Color the unordered pairs of natural numbers with $8$ colors as follows. Suppose $i,jinmathbb N$, $ilt j$. Give the pair ${i,j}$ the color $(x,y,z)$ where $x$ is the truth value of "there is a loop at $v_i$", $y$ is the truth value of "there is a directed edge $v_iv_j$", and $z$ is the truth value of "there is a directed edge $v_jv_i$". Apply Ramsey's theorem to get an infinite monochromatic set. Which of the $8$ colors are possible depends on what you assume about the starting graph.
$endgroup$
– bof
Jan 9 at 12:37
add a comment |
$begingroup$
If every pair of vertices is joined by directed edges in both directions, thhen none of your four alternatives holds. I guess you are assuming some antisymmetry: " if edges $ab$ and $ba$ are both present, then $a=b$" or something like that. I don't think that is the usual meaning of the term "directed graph".
$endgroup$
– bof
Jan 9 at 12:30
$begingroup$
Would such a graph contain an infinite vertex set that would induce an infinite clique?
$endgroup$
– orange
Jan 9 at 12:35
$begingroup$
Here is how to use Ramsey's theorem. Choose an infinite sequence $v_1,v_2,dots$ of distinct vertices. Color the unordered pairs of natural numbers with $8$ colors as follows. Suppose $i,jinmathbb N$, $ilt j$. Give the pair ${i,j}$ the color $(x,y,z)$ where $x$ is the truth value of "there is a loop at $v_i$", $y$ is the truth value of "there is a directed edge $v_iv_j$", and $z$ is the truth value of "there is a directed edge $v_jv_i$". Apply Ramsey's theorem to get an infinite monochromatic set. Which of the $8$ colors are possible depends on what you assume about the starting graph.
$endgroup$
– bof
Jan 9 at 12:37
$begingroup$
If every pair of vertices is joined by directed edges in both directions, thhen none of your four alternatives holds. I guess you are assuming some antisymmetry: " if edges $ab$ and $ba$ are both present, then $a=b$" or something like that. I don't think that is the usual meaning of the term "directed graph".
$endgroup$
– bof
Jan 9 at 12:30
$begingroup$
If every pair of vertices is joined by directed edges in both directions, thhen none of your four alternatives holds. I guess you are assuming some antisymmetry: " if edges $ab$ and $ba$ are both present, then $a=b$" or something like that. I don't think that is the usual meaning of the term "directed graph".
$endgroup$
– bof
Jan 9 at 12:30
$begingroup$
Would such a graph contain an infinite vertex set that would induce an infinite clique?
$endgroup$
– orange
Jan 9 at 12:35
$begingroup$
Would such a graph contain an infinite vertex set that would induce an infinite clique?
$endgroup$
– orange
Jan 9 at 12:35
$begingroup$
Here is how to use Ramsey's theorem. Choose an infinite sequence $v_1,v_2,dots$ of distinct vertices. Color the unordered pairs of natural numbers with $8$ colors as follows. Suppose $i,jinmathbb N$, $ilt j$. Give the pair ${i,j}$ the color $(x,y,z)$ where $x$ is the truth value of "there is a loop at $v_i$", $y$ is the truth value of "there is a directed edge $v_iv_j$", and $z$ is the truth value of "there is a directed edge $v_jv_i$". Apply Ramsey's theorem to get an infinite monochromatic set. Which of the $8$ colors are possible depends on what you assume about the starting graph.
$endgroup$
– bof
Jan 9 at 12:37
$begingroup$
Here is how to use Ramsey's theorem. Choose an infinite sequence $v_1,v_2,dots$ of distinct vertices. Color the unordered pairs of natural numbers with $8$ colors as follows. Suppose $i,jinmathbb N$, $ilt j$. Give the pair ${i,j}$ the color $(x,y,z)$ where $x$ is the truth value of "there is a loop at $v_i$", $y$ is the truth value of "there is a directed edge $v_iv_j$", and $z$ is the truth value of "there is a directed edge $v_jv_i$". Apply Ramsey's theorem to get an infinite monochromatic set. Which of the $8$ colors are possible depends on what you assume about the starting graph.
$endgroup$
– bof
Jan 9 at 12:37
add a comment |
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$begingroup$
If every pair of vertices is joined by directed edges in both directions, thhen none of your four alternatives holds. I guess you are assuming some antisymmetry: " if edges $ab$ and $ba$ are both present, then $a=b$" or something like that. I don't think that is the usual meaning of the term "directed graph".
$endgroup$
– bof
Jan 9 at 12:30
$begingroup$
Would such a graph contain an infinite vertex set that would induce an infinite clique?
$endgroup$
– orange
Jan 9 at 12:35
$begingroup$
Here is how to use Ramsey's theorem. Choose an infinite sequence $v_1,v_2,dots$ of distinct vertices. Color the unordered pairs of natural numbers with $8$ colors as follows. Suppose $i,jinmathbb N$, $ilt j$. Give the pair ${i,j}$ the color $(x,y,z)$ where $x$ is the truth value of "there is a loop at $v_i$", $y$ is the truth value of "there is a directed edge $v_iv_j$", and $z$ is the truth value of "there is a directed edge $v_jv_i$". Apply Ramsey's theorem to get an infinite monochromatic set. Which of the $8$ colors are possible depends on what you assume about the starting graph.
$endgroup$
– bof
Jan 9 at 12:37