4
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If the five 3D coordinates 003 044 330 404 443 are plotted, all the angles happen to be acute, so this is an acute set. It is believed that five points is the largest acute set in 3 dimensions. Also, all the points are on the integer grid, and the largest-smallest grid values is 4-0 or 4, so this acute set has a tightness of 4.

In 4D, the acute set {{8,8,0,0}, {-8,-8,0,0}, {0,8,8,2}, {0,-8,-8,2}, {8,0,8,1}, {-8,0,-8,1}, {6,5,6,8}, {-6,-5,-6,8}} has a tightness of 16. It is believed that 8 points is the largest acute set in 4D.

For real starting values I'm using Viktor Harangi's Acute sets in Euclidean spaces.

In 5D, the largest known acute set has 12 points. A tightness of 228 comes from
{{114,0,0,0,1}, {-114,0,0,0,1}, {0,114,0,0,2}, {0,-114,0,0,2},
{0,0,114,0,3}, {0,0,-114,0,3}, {0,0,0,114,0}, {0,0,0,-114,0},
{31,31,31,31,99}, {-31,-31,-31,-31,99}, {55,55,-55,-55,-30}, {-55,-55,55,55,-30}}

In 6D, the largest known acute set has 16 points.
In 7D, the largest known acute set has 24 points.
In 8D, the largest known acute set has 32 points.
In 9D, the largest known acute set has 48 points.
In 10D, the largest known acute set has 72 points.

What are the tightest presentations for maximal acute sets? Do I even have it right for 3D and 4D?

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    $\begingroup$ By brute force you have it right for 3D; I find 304 solutions for 3D with tightness 4, although that's with double-counting of isometric solutions, and none with tightness 3. $\endgroup$ – Peter Taylor Jul 19 '17 at 11:51
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    $\begingroup$ I have a better example for 4D and 8 points: {(0, 0, 1, 0) (0, 7, 1, 1) (7, 1, 0, 2) (5, 6, 2, 7) (7, 7, 6, 0) (7, 0, 6, 1) (0, 6, 7, 2) (2, 1, 5, 7)}. It is optimal tight, I think. I also have examples for 4D-9 points and 5D-17 points. I'll post them later. $\endgroup$ – grizzly Sep 13 '17 at 15:59
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Recently, new results have been obtained for this problem.

First, D. Zakharov constructed an acute set in $\mathbb R^d$ of size at least $1.618^d$.

Thereafter Balázs Gerencsér and Viktor Harangi presented a simple construction of an acute set of size $2^{d−1}+1$ in $\mathbb R^d$ for any dimension $d$.

My examples for maximal accute sets:

Example for 4D-9 points:

(0, 1, 2, 0)  (0, 42, 1, 1)  (42, 1, 0, 2)  (41, 41, 0, 13) 
(44, 43, 42, 0)  (44, 2, 43, 1)  (2, 43, 44, 2)  (3, 3, 44, 13) 
(32, 32, 15, 44)

I think that this is not the optimal example, but almost.

Example for 5D-17 points (this example is not optimal, I'm sure :)

(6000, 6000, 6000, 6000, 3100000) 
(199999776, 100, 300, 240, 196000) 
(14, 199999968, 0, 2, 601998) 
(200000000, 199999986, 0, 0, 695818) 
(0, 0, 199999986, 0, 695832) 
(200000000, 2, 200000000, 12, 695840) 
(0, 199999997, 199999984, 0, 727498) 
(199999998, 200000000, 199999988, 0, 695858) 

(199994000, 199994000, 199994000, 199994000, 3100000) 
(224, 199999900, 199999700, 199999760, 196000) 
(199999986, 32, 200000000, 199999998, 601998) 
(0, 14, 200000000, 200000000, 695818) 
(200000000, 200000000, 14, 200000000, 695832) 
(0, 199999998, 0, 199999988, 695840) 
(200000000, 3, 16, 200000000, 727498) 
(2, 0, 12, 200000000, 695858) 

(120000000, 120000000, 120000000, 120000000, 200000000)

Upd. 6.10.2017 I improved my understanding and algorithms for 4D-9 points:

(1, 5, 5, 9)  (2, 22, 4, 1)  (20, 2, 2, 0)  (22, 20, 2, 2)  
(23, 19, 19, 9)  (22, 2, 20, 1)  (4, 22, 22, 0)  (2, 4, 22, 2)  
(13, 12, 12, 24)
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I found a slightly smaller solution for 9 points in 4D:

(1 10 3 8)   (12 19 1 9) (1 15 18 7) (10 1 7 0)
(21 12 19 8) (10 3 21 9) (21 7 4 7)  (12 21 15 0)
(11 11 11 22) 
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  • $\begingroup$ Whoops, no... {{10, 3, 8}, {3, 21, 9}, {7, 4, 7}} is not an acute triangle. $\endgroup$ – Ed Pegg Jan 9 '18 at 17:30
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    $\begingroup$ @EdPegg This solution is correct. It is 4D, not 3D :) $\endgroup$ – grizzly Jan 9 '18 at 17:54
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grizzly and I have found a significantly smaller solution for 17 points in 5D with tightness 13300. Attempts to reduce this solution further were not successful and a new approach is needed.

(2862 9245 12725 10241 9223)
(9044 1299 1136 9699 9301)
(10885 9549 10954 9534 9280)
(9794 1724 8486 0 9631)
(8092 11963 1334 3000 9491)
(9090 10522 2490 11056 8293)
(9450 10593 9550 1975 13300)
(71 11657 3105 3716 9698)
(8023 2718 0 1580 9531)
(1841 10664 11589 2126 9565)
(0 2414 1771 2289 9530)
(1091 10239 4239 11824 9207)
(2793 0 11391 8822 9291)
(1796 1439 10237 700 8687)
(1443 1384 3165 10124 13022)
(10812 305 9620 8123 9560)
(5951 7380 7313 5115 0)

Update 6/05/2018. We managed to improve our result to tightness 8564:

(5632 0 4 6 5743)
(5632 5635 2 5 5748)
(0 1 2 5629 5740)
(5 5635 2 0 5741)
(5 3 5636 2 5742)
(5614 13 15 5622 6278)
(262 256 260 256 8564)
(5631 3 5635 8 5610)
(0 5638 5632 5630 5743)
(0 3 5634 5631 5748)
(5632 5637 5634 7 5740)
(5627 3 5634 5636 5741)
(5627 5635 0 5634 5742)
(18 5625 5621 14 6278)
(5370 5382 5376 5380 8564)
(1 5635 1 5628 5610)
(2593 2582 2582 2588 0)
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