Timeline for Ramsey-Turán density function is well defined
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Oct 8, 2021 at 5:11 | comment | added | domotorp | @JPM No, that's not true for any sequence. Here we should use that the sequence $a_n$ is almost continuous, so something like $a_k$ and $a_{k+1}$ are not far, though even this does not seem to be trivial to prove, I'm positive this approach works. Regarding $L^{(r)}$, I don't see what to do. | |
| Oct 8, 2021 at 3:02 | comment | added | JPMarciano | It is also not completely clear on how to finish it. This proves that $ex_l(n,\alpha n) \leq k^2ex_l(kn, \alpha (kn))$ for every $k$ and $n$. That is, $$\frac{ex_l(n,\alpha n)}{n^2}\leq \frac{ex_l(nk,\alpha (nk))}{(nk)^2}.$$ But how to finish with it? Exactly the Fekete Lemma doesn't seem to fit. Is it true that any limited sequence $(a_n)_{n \in\mathbb{N}}$ satisfying $a_{n} \leq a_{nk}$ for every $k,n \in \mathbb{N}$ converges? | |
| Oct 7, 2021 at 21:23 | comment | added | JPMarciano | This $L^{(r)}$ in Simonovits and Sós survey is a general $r$-uniform hypergraph. | |
| Oct 7, 2021 at 21:18 | comment | added | domotorp | Why do you think that it holds for general graphs? | |
| Oct 7, 2021 at 21:16 | comment | added | JPMarciano | Ok, thanks a lot! One more thing, for cliques it is clear that this new blow-up graph will be $K_l$-free. How is this true for a general graph? For example, $K_{2,2,2} \not \subset K_3$, but applying your idea in $K_3$ with $k \geq 2$ would imply that our new graph has a $K_{2,2,2}.$ | |
| S Oct 7, 2021 at 21:16 | history | suggested | JPMarciano | CC BY-SA 4.0 |
vertex instead of edge
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| Oct 7, 2021 at 21:09 | review | Suggested edits | |||
| S Oct 7, 2021 at 21:16 | |||||
| Oct 7, 2021 at 20:53 | comment | added | domotorp | @JPM No, just blowing up vertices. For example, if your graph is just a $K_2$, the edge $uv$, then if $k=2$, the new graph becomes a $C_4$, $v_1u_1v_2u_2$. | |
| Oct 7, 2021 at 20:16 | comment | added | JPMarciano | Can you explain this idea a little bit better? Are you deleting the edge and adding vertices? | |
| Oct 7, 2021 at 19:06 | comment | added | domotorp | @JPM You are right, this follows from vertex-multiplication. | |
| Oct 7, 2021 at 19:06 | history | edited | domotorp | CC BY-SA 4.0 |
Fekete lemma
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| Oct 7, 2021 at 16:37 | comment | added | JPMarciano | I believe that in this case it would be quite surprising if it is just a definition problem. Lüders, Reiher (2017) say it is well known and easy to confirm that this limit exists. Łuczak, Polcyn, Reiher (2020) also say that this limit exists, referring to Erdos, Hajnal, Sós, Szemerédi (1983), but I couldn't find it there. | |
| Oct 7, 2021 at 6:15 | history | answered | domotorp | CC BY-SA 4.0 |