Timeline for How many arrangements of $n$ points with $k$ edge lengths exist in $d$ dimensions?
Current License: CC BY-SA 4.0
7 events
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| Nov 15, 2019 at 18:39 | history | edited | Benjamin Dickman | CC BY-SA 4.0 |
Broadening the scope of welcome responses after receiving 0 so far
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| Nov 8, 2019 at 1:59 | comment | added | Behnam Esmayli | SHouldn't there be a definition of being "the same configuration" be included in the question? | |
| Nov 7, 2019 at 21:33 | comment | added | Benjamin Dickman | @GerryMyerson This pointer is great; thanks! Alon Amit's response from just four days ago - and using Desmos! - is especially timely. Part of our class is focused on developing problems of the right size, and the main question asked here is Way Too Big, So, references that help indicate oversize are of definite interest, as are references that point to the boundaries of known results. | |
| Nov 7, 2019 at 21:13 | comment | added | Gerry Myerson | There are some very hard, even unsolved, questions here. You might enjoy mathoverflow.net/questions/58203/erdos-distance-problem-n-12 See also oeis.org/A131628 and oeis.org/A186704 | |
| Nov 7, 2019 at 20:20 | comment | added | Benjamin Dickman | @WillBrian [Pointing out the obvious but] There are also uninteresting cases when $k > n(n-1)/2 =$ the number of point pairings; specifically, the answer in these cases will be zero: e.g., there is no way to place $4$ points so that exactly $7$ distinct distances arise. I'm not sure whether to include these constraints in the original post; feel free to edit as you deem appropriate! | |
| Nov 7, 2019 at 20:12 | comment | added | Will Brian | One observation: If $d$ is big enough (I think $\geq\! n-1$ is big enough), then all the geometric obstructions disappear and your problem becomes purely combinatorial. Specifically, if $d \geq n-1$ then $(n,k,d)$ yields the answer of however many ways there are to color the edges of $K_n$ with exactly $k$ colors. For example, in $3$ or more dimensions the problem with $4$ points and $2$ distances has $9$ solutions instead of just $6$. So I think the interesting cases are where $1 < d < n-1$, because that's where the geometry really matters. | |
| Nov 7, 2019 at 18:47 | history | asked | Benjamin Dickman | CC BY-SA 4.0 |