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A089676
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a(n) is the maximal size of a set S of points in {0,1}^n in real n-dimensional Euclidean space such that every angle determined by three points in S is acute.
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2
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1, 2, 2, 4, 5, 6, 8, 9, 10, 16, 17
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OFFSET
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0,2
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COMMENTS
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Consider the 2^n points {0,1}^n in real Euclidean space. Then a(n) = maximal size of a subset S of these 2^n points such that there is no triple of points P,Q,R in S which subtends a right angle. That is, we are not allowed to have P-Q perpendicular to R-Q.
There is an existence proof due to Erdős and Füredi that exponentially large subsets S exist: see for example Theorem 2.3 of Noga Alon's survey "Probabilistic Methods in Extremal Finite Set Theory". This was improved by Bevan and later by Ackerman and Ben-Zwi.
As explained by Erdős-Furedi, these sets of points are equivalent to set systems none of which is sandwiched between the intersection and the union of two others. In turn these are equivalent to so-called (2,1)-separating systems. As far as I know the best construction is the one in my Israel J. Math. 2013 paper. It uses algebraic geometry and coding techniques, and it implies a lower bound on a(n) of roughly 11^{3n/50}. This is better than Erdős-Furedi, Bevan, or Ackerman-Ben-Zwi, which are all about (4/3)^{n/2}. It is remarkable that this construction beats the probabilistic method (and note that it implies that for large n, naive computer search will have exponentially small chance to find optimal configurations). I should also add that Erdős-Furedi claimed (without proof) a lower bound of 2^{n/4} which, if correct, would be even better (but also non-constructive). - Hugues Randriambololona, Apr 08 2016
For a(10)=17 a combinatorial search algorithm shows that a cubic acute 10-set with 18 vertices is not possible. For a complete enumeration of sets with maximal size, see A289972. - Fausto A. C. Cariboni, Jul 17 2017
The best known lower bounds for a(11-15) are 24, 32, 33, 64 and 128. a(11-14) were found by D. Kamenetsky, while a(15) was found by D. Kamenetsky and V. Chubenko (see attached file). Lower bounds for n > 15 have been found by V. Harangi (see Table 3 in his paper). - Dmitry Kamenetsky, May 18 2018 and Jun 05 2018
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LINKS
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FORMULA
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If k <= m <= n, a(k+2m) >= a(k)a(m), a(k+2m+3n) >= a(k)a(m)a(n).
a(n) >= 2*floor((sqrt(6)/9)(2/sqrt(3))^n), which is approximately 0.544*1.155^n.
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EXAMPLE
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a(3) = 4: {000, 011, 101, 110}.
a(4) = 5: {0011, 0101, 0110, 1000, 1111}.
The following sets are given by Bevan (2006), who also shows they are optimal:
a(5) = 6:
0 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 1 0 0 1
1 0 0 0 1
1 1 1 1 0
a(6) = 8:
0 0 0 0 0 0
0 0 0 1 1 1
0 1 1 0 0 1
0 1 1 1 1 0
1 0 1 0 1 0
1 0 1 1 0 1
1 1 0 0 1 1
1 1 0 1 0 0
a(7) = 9:
0 0 0 0 0 0 0
0 0 0 0 0 1 1
0 0 0 1 1 0 1
0 1 1 0 0 0 1
0 1 1 1 1 1 0
1 0 1 0 1 0 1
1 0 1 1 0 1 0
1 1 0 0 1 1 0
1 1 0 1 0 0 1
a(8) = 10:
0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1
0 0 0 0 0 1 0 1
0 0 0 1 1 0 0 1
0 1 1 0 0 0 0 1
0 1 1 1 1 1 1 0
1 0 1 0 1 0 0 1
1 0 1 1 0 1 1 0
1 1 0 0 1 1 1 0
1 1 0 1 0 0 0 1
For a(9) = 16 Bevan uses the construction in his Theorem 4.2, which shows that a(3k) >= a(k)^2 for all k, and then a computer search shows that this is optimal for k = 3. Let v0,v1,v2,v3 denote the four vectors for a(3). Then to get a(9)=16 use the vectors { v_i v_j v_{j-i mod 4}, 0 <= i,j <= 3 }. - N. J. A. Sloane, Mar 30 2016
a(10) = 17, from Hugues Randriambololona, Apr 08 2016:
0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 1 0 1 1 1
1 1 1 0 1 1 1 0 0 1
0 0 1 0 0 1 1 1 0 1
1 1 0 0 0 1 0 1 0 1
1 1 0 0 1 0 1 1 1 1
0 1 0 1 1 0 0 1 0 0
1 1 1 0 0 0 0 0 1 1
1 0 0 1 1 1 1 0 1 1
1 0 1 0 1 1 0 0 1 0
0 1 0 1 0 1 1 0 0 0
1 0 0 0 0 1 1 1 1 0
1 0 1 0 1 0 1 1 0 0
1 0 1 1 0 0 1 1 1 1
1 1 1 1 0 1 0 1 1 0
0 1 1 0 1 1 1 1 1 0
1 0 1 1 1 1 0 1 0 1
0 1 1 0 0 0 1 0 1 1 1
0 0 0 0 1 0 0 0 0 0 1
0 0 1 1 0 0 1 0 0 0 1
0 1 1 0 1 0 1 1 1 0 0
0 0 1 0 1 1 0 0 1 1 1
1 1 0 1 1 0 0 0 1 1 0
1 0 0 1 1 0 0 1 0 1 1
1 0 0 1 0 1 1 0 1 1 0
1 1 1 1 1 1 0 0 0 0 0
1 0 0 1 1 1 1 1 1 0 1
1 0 1 0 0 0 0 0 0 0 0
1 0 0 0 0 1 0 1 1 1 1
0 1 0 0 1 0 1 0 0 1 0
1 1 0 0 0 0 0 0 1 0 1
1 1 1 0 0 1 1 1 0 1 1
1 1 0 0 0 1 1 0 0 0 0
0 0 0 1 0 0 1 1 0 1 0
1 0 1 1 1 1 1 0 0 1 1
1 1 1 1 0 0 0 1 0 0 1
0 1 1 1 0 0 0 1 1 1 0
0 1 1 0 1 1 0 1 0 1 0
0 0 0 0 0 1 0 1 0 0 0
0 1 0 1 0 1 0 0 0 0 1
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CROSSREFS
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KEYWORD
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nonn,more,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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