Timeline for Does every bijective graph endomorphism restrict to a full-cardinality isomorphism?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 10, 2019 at 20:22 | comment | added | Alex Meiburg | Mm, right, makes sense now. Thank you! | |
| Feb 10, 2019 at 20:17 | comment | added | Ramiro de la Vega | @AlexMeiburg, Sorry if I was too vague. Todorcevic showed that for any successor cardinal $\kappa$ there exists a coloring $c:[\kappa]^2 \to \omega$ such that for any $A \subseteq \kappa$ of size $\kappa$, the restriction of $c$ to $A$ takes all possible colors. The structure of the set of colors is irrelevant so we can always make it $\mathbb{Z}$ which allows us to construct $G$ and $f$ just as we did with $\omega_1$. | |
| Feb 10, 2019 at 19:52 | vote | accept | Alex Meiburg | ||
| Feb 10, 2019 at 19:52 | comment | added | Alex Meiburg | Wow! What an amazingly unexpected answer! I absolutely did not expect it to depend so sensitively on the cardinality. To check that I understand: if Q is the successor cardinal of P, the Todorcevic construction uses $V(G)=Q\times P$? It's known that such a $P$-coloring of $K_Q$ will always exist? The mapping $f$ requires ordering $P$ as $P$ successive copies of $\mathbb{Z}$? Thank you! | |
| Feb 10, 2019 at 16:15 | history | edited | Ramiro de la Vega | CC BY-SA 4.0 |
added 315 characters in body
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| Feb 10, 2019 at 16:06 | history | answered | Ramiro de la Vega | CC BY-SA 4.0 |