Timeline for Ramsey-Kuratowski numbers
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Jun 9, 2023 at 10:04 | history | edited | Tony Huynh | CC BY-SA 4.0 |
added 1 character in body; edited tags
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| Jun 9, 2023 at 9:58 | answer | added | Tony Huynh | timeline score: 2 | |
| Nov 22, 2019 at 17:17 | comment | added | Taras Banakh | @WillBrian Oh, sorry! I made a mistake thinking that there are two triangles (with vertices $(1,i)$ and $(2,i)$ for $i\in\{1,2,3\}$, but there are edges between these triabgkes, so they are not disjoint). | |
| Nov 22, 2019 at 13:10 | comment | added | Will Brian | @TarasBanakh: I cannot find the copy of $K_{3,3}$ you're referring to. Would you mind pointing it out more explicitly, perhaps by listing the coordinates? | |
| Nov 22, 2019 at 5:02 | comment | added | Wlod AA | I hope that I clarified my Question. (Did I?) | |
| Nov 22, 2019 at 5:01 | history | edited | Wlod AA | CC BY-SA 4.0 |
Clearer (?)
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| Nov 21, 2019 at 23:56 | comment | added | Wlod AA | @TarasBanakh, thank you -- done. | |
| Nov 21, 2019 at 23:52 | history | edited | Wlod AA | CC BY-SA 4.0 |
A quote from Taras Banakh
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| Nov 21, 2019 at 3:24 | comment | added | Taras Banakh | @WlodAA Of course, you can! | |
| Nov 21, 2019 at 0:41 | comment | added | Wlod AA | Everybody who has partial answers, I encourage you to post them as Answers. This way you will stimulate the progress -- in real-time -- for the given topic. Somehow, one reacts to Answers stronger than to comments. | |
| Nov 21, 2019 at 0:37 | comment | added | Wlod AA | @TarasBanakh, may I quote your useful comment about $R(5\,5)$ in my Question ? | |
| Nov 20, 2019 at 17:16 | comment | added | Wlod AA | I have to rush now, am about to play in the last round of a local senior chess tournament (I am 3:0 and will play against the other highest scorer, who has 2.5pts). I'll see you later. | |
| Nov 20, 2019 at 14:33 | comment | added | Will Brian | It's not enough for an answer, but I can show $13 \leq \mathrm{RK}_{xx}$. To see this, we need a $2$-coloring of the edges of $K_{12}$ that avoids any monochromatic copies of $K_5$ or $K_{3,3}$. To get one, just label the vertices with ordered pairs $(a,b)$ such that $a \in \{1,2,3\}$ and $b \in \{1,2,3,4\}$, and then put an edge between two vertices iff either their first coordinates are equal or their second coordinates are equal. | |
| Nov 20, 2019 at 14:14 | comment | added | Fedor Petrov | @TarasBanakh why should it be the same? Non-planarity is implied by a homeomorphic copy of $K_5$ or $K_{3,3}$, not isomorphic. | |
| Nov 20, 2019 at 13:45 | comment | added | Taras Banakh | @WlodAA What about this form of the question: find the smallest $n$ such that for any graph $\Gamma$ on $n$ vertices either $\Gamma$ or the complement of $\Gamma$ in $K_n$ is non-planar. Is this the same as your $RK_{xx}$? | |
| Nov 20, 2019 at 13:33 | comment | added | Taras Banakh | @WlodAA The problem with the current formulation is that if you have a monochrome $K_6$, then you also have a monochrome $K_{3,3}$. But I suggest that asking the question you had in mind something different: there exists a copy of $K_6$ whose red colored edges (say) form a subgraph isomorphic to $K_{3,3}$ (and then green colored edges form two disjoint copies of $K_3$). So, which version of the question had you in mind? | |
| Nov 20, 2019 at 8:57 | comment | added | Wlod AA | @IlyaBogdanov, I mean that each of the graphs $\,K_{3\,3}\, $ and $\,K_5\,$ should have edges of one color (all $9$ edges of $\,K_{3\,3}\,$ should be of the same color). How should I write it more clearly? You're welcome to try your hand at editing my Q. | |
| Nov 20, 2019 at 8:46 | comment | added | Ilya Bogdanov | I think @Taras's question addresses the word "induce". When you get a monocolor $K_{3,3}$, do you insist to have all other edges between the six vertices to have the other color? If not, it is better to replace 'induce' by some other word. | |
| Nov 20, 2019 at 8:27 | comment | added | Wlod AA | @TarasBanakh, your comment about $K_{3,3}$, I don't understand (of course edges of $\,K_6\,$ cannot be two-colored). | |
| Nov 20, 2019 at 8:08 | comment | added | Taras Banakh | Observe that $RK_{xx}\le R(5,5)\le 48. The is a conjecture that R(5,5)=43. | |
| Nov 20, 2019 at 7:45 | comment | added | Taras Banakh | I hope the graph $K_{3,3}$ should be defined as a suitably two-colored $K_6$. | |
| Nov 19, 2019 at 22:49 | history | edited | Wlod AA | CC BY-SA 4.0 |
an inequality
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| Nov 19, 2019 at 22:42 | history | edited | Wlod AA | CC BY-SA 4.0 |
I missed an explanation
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| Nov 19, 2019 at 21:20 | history | asked | Wlod AA | CC BY-SA 4.0 |