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Szemeredi's regularity lemma for countably infinite graphs?
Mentally cross off, "so the generalization", in the comment beginning with, "I see, thanks." Thanks.
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Szemeredi's regularity lemma for countably infinite graphs?
(cont) nonstandard integers in $PA$ (after all, what nonstandard integer would 1+2+3+4+..., be, anyway)?
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Szemeredi's regularity lemma for countably infinite graphs?
I see, thanks. Is there a way to generalize the notion of epsilon-regularity so that finite graphs would participate in (so to speak) in being epsilon-regular and the countably infinite graphs would participate in the generalization so the generalization so the Szemeredi Regularity lemma would hold for finite graphs and its generalization (with the generalization of epsilon-regularity) would hold for countably infinite graphs (one should not be expected that the aforementioned generalization should hold for countably infinite graphs since there is no relation between standard integers and
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Szemeredi's regularity lemma for countably infinite graphs?
(cont.) "The only countable partition-regular graphs are the complete graph, the null graph, and $R$, namely, only these three graphs are such that any finite vertex coloring yields a color whose induced subgraph is isomorphic to the original graph"? For a countably infinite graph, wouldn't that qualitative property suffice to define regularity ( and couldn't one express Szemeredi's Regularity Lemma qualitatively in terms of vertex colorings or have I misunderstood the term "partition-regular")?
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Szemeredi's regularity lemma for countably infinite graphs?
Well, isn't the Rado graph universal with respect to this property?
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Szemeredi's regularity lemma for countably infinite graphs?
@TerryTao: Regarding my question, would it make sense to try to apply Szmeredi's Regularity Lemma to the Rado graph $R$, since all finite and countably infinite graphs are induced subgraphs of $R$?
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Szemeredi's regularity lemma for countably infinite graphs?
@TerryTao: Thanks.
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Szemeredi's regularity lemma for countably infinite graphs?
@TerryTao: any references regarding the Banach means you are referring to?
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Szemeredi's regularity lemma for countably infinite graphs?
@TerryTao: thank you. Very helpful. Possibly silly question: if one were to formulate the infinite measure in nonstandard analysis, would that help?
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Request for references to literature: quantifying the complexity of cardinal arithmetic
What type of complexity hierarchy/analogue of recursiveness for functions on cardinals would you be hoping to find?