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Assume $e$ is any edge in a graph $R$ and $v$ is any vertex in a graph $R$.

  1. Show that $R-e$ is a Rado Graph if and only if $R$ is a Rado Graph.
  2. Show that $R-v$ is a Rado Graph if $R$ is a Rado Graph

Do these proofs look good?

(1)

For the removal of an edge $e$, the proof follows from considering the Rado property that $\forall$ 2 finite sets of vertices $V, W \in R$, $\exists$ a vertex $v$ that is connected by edges to every vertex in $V$ and not connected to any vertex in $W$.

Assume that $e$ was one of these edges joining a vertex $x \in V$ to a vertex $v \in (V \cup W) \backslash R$ satisfying the Rado property for the given $V, W$. I may also choose $V'$ and $W'$ to be sets s.t. $x \in W$ and $V' \cup W' = V \cup W$. Then still $\exists$ $v' \in (V \cup W) \backslash R$ satisfying the Rado property for the given $V', W' \in R$. Now, in the graph $R-e$, $v'$ will satisfy the Rado property for $V, W$, and $v$ will satisfy the Rado property for $V', W'$. The argument for the only if part of this condition is the same but consider adding an edge to the graph $R-e$. $\square$

(2)

If I remove a vertex $x$ from the graph $R$, I may well have first removed $deg(x)$ edges from $x$. The Rado property will hold under removal of $deg(x)$ edges from a vertex $x \in R$ by repeated application of (1). Now I want to remove $x$ which should cause no issue given that x could only be the vertex satisfying the Rado property for $R$ given $V=\phi$ and $W$ finite with $W \in R-v$, but $R$ was a Rado graph before removing all the edges from $v$ (which would have connected to some vertices possibly in $W$), so there is a vertex $v \neq x \in R$ that will satisfy the Rado property in this case. Thus, $R-v$ is necessarily Rado. $\square$

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    $\begingroup$ Does it have to be that complicated? Don't we know that, if $V$ and $W$ are disjoint finite sets of vertices in the Rado graph, than there is not just one but an infinite number of vertices $x$ joined to everything in $V$ and nothing in $W$? And isn't it pretty clear than adding or subtracting an edge, or subtracting a vertex, can't spoil more than $1$ of those infinitely many $x$'s? $\endgroup$ – bof Dec 2 '18 at 11:55
  • $\begingroup$ If we suppose that the Rado property is satisfied by more than one vertex for any 2 finite sets then yes, but the definition doesn't tell me i get 2, just at least 1. So yes, i should have to care about there possibly being only one vertex satisfying the property for given V and W $\endgroup$ – rjm27trekkie Dec 2 '18 at 16:45
  • $\begingroup$ Then why don't you first prove as a lemma that you can find more than one vertex for given $V$ and $W$? (Hint: what if you take the first vertex $x$ that you got, add it to $V$, and use the Rado property again?) $\endgroup$ – bof Dec 3 '18 at 11:25
  • $\begingroup$ By the way, I believe the usual definition of the Rado property says that there is a vertex $x$ such that $x$ is joined to every vertex in $V$ and to no vertex in $W$, and $x\notin W$. You left out the $x\notin W$ part. Your version works too, but makes slightly more work. You might start by proving that your version is equivalent to the usual one. $\endgroup$ – bof Dec 3 '18 at 11:28
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I disagree with part of your first proof. A vertex $v \in (V \cup W) \backslash R$ makes no sense, since $V$ and $W$ are both subsets of R, thus $(V \cup W) \backslash R = \phi$. A definition of $R \backslash (V \cup W)$ makes more sense to me, especially given how you defined the Rado property (each vertex $v$ which fulfills the property for some $V$ and $W$ cannot be a member of either $V$ nor $W$).

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  • $\begingroup$ This is the definition. $V \subset R, W \subset R$ but $V \cup W \neq R$. They are not spanning subsets of $R$ because they are FINITE while $R$ is countably infinite. (V \cup W) | R describes all these vertices not in V or W (2 finite subsets of R) that are also in R (that's what the relative set complement under R (\ operator) means $\endgroup$ – rjm27trekkie Dec 3 '18 at 15:38
  • $\begingroup$ en.wikipedia.org/wiki/Complement_(set_theory) I stand by the statement that $(V \cup W) \backslash R$ means all members of $V \cup W$ which are not in $R$. $\endgroup$ – theasianpianist Dec 3 '18 at 15:54

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