Timeline for a Ramsey-type question
Current License: CC BY-SA 3.0
20 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Jul 13, 2011 at 20:16 | comment | added | gowers | Yes it was then. | |
| Jul 13, 2011 at 15:37 | comment | added | user6976 | I know that Noga Alon became interested in these questions (after my talk at a conference in Cambridge last January). | |
| Jul 13, 2011 at 14:08 | comment | added | gowers | By the way, I find this cluster of related questions you are asking very appealing. (My attention was first drawn to them by Noga Alon.) | |
| Jul 13, 2011 at 14:07 | comment | added | gowers | Oops, I meant category. The rough idea I had in mind was that if you cover the $n$-sphere with $m$ open sets, then it would be nice to show that one of them was non-trivial in some topological sense. The trouble is, they're all contractible, but perhaps one could hope that one of them intersected with minus itself is not contractible when considered as a subset of projective $n$-space. That's not quite the same as Lusternik-Schnirelman category but it seems to be in a similar ball park and I don't rule out some way of getting from one to the other. | |
| Jul 12, 2011 at 22:39 | comment | added | user6976 | @Gerhard: He considers the same graph: vertices are subsets of $S$ with two subsets connected if their symmetric difference has at most 2 element. So it may be related indeed. Thanks! | |
| Jul 12, 2011 at 22:26 | comment | added | Gerhard Paseman | Found it. The poster mentions Sapozhenko. I am less sure how applicable this problem is, but it may still be of interest. Gerhard "Another Load Off My Mind" Paseman, 2011.07.12 | |
| Jul 12, 2011 at 22:07 | comment | added | user6976 | What is "Lusternik-Schnirelmann capacity"? I thought that Lusternik-Schnirelmann category is about covers by open sets. The continuous version does make sense of course. In fact an interesting problem is about coloring of $\mathbb{R}^k$ and existence of monochromatic 2-paths (in the $l_1$-metric) of arbitrary diameter. | |
| Jul 12, 2011 at 21:23 | comment | added | gowers | What's known about the continuous version? That is, if you cover the sphere $S_n$ with m closed sets, what can be said about the structures that must be contained in at least one of the sets? If m=n+1 then we get two antipodal points, but what if m=1000? It seems that one ought to get much more. At the moment I don't even see a counterexample to the assertion that if m=n then you get a path from a point to the antipodal point, though that seems a bit optimistic. Am I asking about the Lusternik-Schnirelmann capacity of projective space or something like that? | |
| Jul 12, 2011 at 21:15 | comment | added | Gerhard Paseman | My memory says this question that I am trying to remember was not one of yours, It was trying to compute something like the expected size of a 2 neighborhood of a graph, and had a reference whose name I am also trying to remember. Unless this triggers someone else's memory, I'n afraid we will have to let me continue the struggle. Gerhard "Connected Associative Memories Work Better" Paseman, 2011.07.12 | |
| Jul 12, 2011 at 20:54 | comment | added | user6976 | @Gerhard: These were my questions, I refer to them here. | |
| Jul 12, 2011 at 20:25 | answer | added | Tony Huynh | timeline score: 3 | |
| Jul 12, 2011 at 20:20 | history | edited | user6976 | CC BY-SA 3.0 |
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| Jul 12, 2011 at 20:18 | comment | added | Gerhard Paseman | I'll take that as meaning that both versions are of interest. I am struggling to remember a MathOverflow question which used the term "2-connected" to mean roughly that points were within distance 2 of one another. Although the enumeration problem discussed was different, you might still find it of interest in tackling this problem. When I find it, I'll post a link to it. Gerhard "Email Me About System Design" Paseman, 2011.07.12 | |
| Jul 12, 2011 at 19:43 | history | edited | user6976 | CC BY-SA 3.0 |
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| Jul 12, 2011 at 19:32 | comment | added | user6976 | @Gerhard: You are correct. | |
| Jul 12, 2011 at 19:31 | history | edited | user6976 | CC BY-SA 3.0 |
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| Jul 12, 2011 at 19:24 | answer | added | Fedor Petrov | timeline score: 2 | |
| Jul 12, 2011 at 19:12 | comment | added | Gerhard Paseman | A natural assumption is that the A_i are all required to be distinct. Another is if A_i = A_j, then either i=j or A_i+1 is not equal to A_j+1. Are you interested in no cycles or in a long Eulerian path? Gerhard "Email Me About System Design" Paseman, 2011.07.12 | |
| Jul 12, 2011 at 17:52 | history | asked | user6976 | CC BY-SA 3.0 |