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%I #104 Nov 12 2018 07:35:26
%S 1,2,2,4,5,6,8,9,10,16,17
%N 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.
%C 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.
%C 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.
%C 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
%C 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
%C 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
%H Eyal Ackerman and Oren Ben-Zwi, <a href="https://doi.org/10.1016/j.ejc.2008.07.020">On sets of points that determine only acute angles</a>, European Journal of Combinatorics 30 (2009) 908-910.
%H N. Alon, <a href="http://www.cs.tau.ac.il/~nogaa/PDFS/set3.pdf">Probabilistic Methods in Extremal Finite Set Theory</a>
%H D. Bevan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1r12">Sets of Points Determining Only Acute Angles and Some Related Coloring Problems</a>, Electronic J. of Combinatorics, 13(1), 2006, #R12.
%H L. Danzer and B. Grünbaum, <a href="http://dx.doi.org/10.1007/BF01193107">Uber zwei Probleme bezüglich konvexer Körper von P. Erdős und von K. L. Klee</a>, Math. Zeitschrift 79 (1962) 95-99.
%H P. Erdős and Z. Füredi, <a href="http://dx.doi.org/10.1016/S0304-0208(08)73398-X">The greatest angle among n points in the d-dimensional Euclidean space</a>, Annals of Discrete Math. 17 (1983) 275-283.
%H Viktor Harangi, <a href="https://users.renyi.hu/~harangi/papers/acute.pdf">Acute Sets In Euclidean Spaces</a>, Journal on Discrete Mathematics, volume 25, issue 3, (2011) 1212-1229.
%H Dmitry Kamenetsky, <a href="/A089676/a089676_1.txt">Lower bounds and their solutions for a(11-15)</a>
%H Math StackExchange, <a href="https://math.stackexchange.com/questions/2363546/tight-acute-sets">Tight Acute Sets</a>
%H Hugues Randriambololona, <a href="https://doi.org/10.1007/s11856-012-0126-9">(2,1)-Separating systems beyond the probabilistic bound</a>, Israel Journal of Mathematics, 195 (2013), 171-186; DOI: 10.1007/s11856-012-0126-9.
%H Hugues Randriambololona, <a href="https://arxiv.org/abs/1010.5764">(2,1)-Separating systems beyond the probabilistic bound</a>, arXiv:1010.5764 [math.CO], 2010-2012.
%F If k <= m <= n, a(k+2m) >= a(k)a(m), a(k+2m+3n) >= a(k)a(m)a(n).
%F a(n) >= 2*floor((sqrt(6)/9)(2/sqrt(3))^n), which is approximately 0.544*1.155^n.
%e a(3) = 4: {000, 011, 101, 110}.
%e a(4) = 5: {0011, 0101, 0110, 1000, 1111}.
%e The following sets are given by Bevan (2006), who also shows they are optimal:
%e a(5) = 6:
%e 0 0 0 0 0
%e 0 0 0 1 1
%e 0 0 1 0 1
%e 0 1 0 0 1
%e 1 0 0 0 1
%e 1 1 1 1 0
%e a(6) = 8:
%e 0 0 0 0 0 0
%e 0 0 0 1 1 1
%e 0 1 1 0 0 1
%e 0 1 1 1 1 0
%e 1 0 1 0 1 0
%e 1 0 1 1 0 1
%e 1 1 0 0 1 1
%e 1 1 0 1 0 0
%e a(7) = 9:
%e 0 0 0 0 0 0 0
%e 0 0 0 0 0 1 1
%e 0 0 0 1 1 0 1
%e 0 1 1 0 0 0 1
%e 0 1 1 1 1 1 0
%e 1 0 1 0 1 0 1
%e 1 0 1 1 0 1 0
%e 1 1 0 0 1 1 0
%e 1 1 0 1 0 0 1
%e a(8) = 10:
%e 0 0 0 0 0 0 0 0
%e 0 0 0 0 0 0 1 1
%e 0 0 0 0 0 1 0 1
%e 0 0 0 1 1 0 0 1
%e 0 1 1 0 0 0 0 1
%e 0 1 1 1 1 1 1 0
%e 1 0 1 0 1 0 0 1
%e 1 0 1 1 0 1 1 0
%e 1 1 0 0 1 1 1 0
%e 1 1 0 1 0 0 0 1
%e 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
%e a(10) = 17, from Hugues Randriambololona, Apr 08 2016:
%e 0 0 0 0 0 0 0 0 0 0
%e 0 0 0 0 1 1 0 1 1 1
%e 1 1 1 0 1 1 1 0 0 1
%e 0 0 1 0 0 1 1 1 0 1
%e 1 1 0 0 0 1 0 1 0 1
%e 1 1 0 0 1 0 1 1 1 1
%e 0 1 0 1 1 0 0 1 0 0
%e 1 1 1 0 0 0 0 0 1 1
%e 1 0 0 1 1 1 1 0 1 1
%e 1 0 1 0 1 1 0 0 1 0
%e 0 1 0 1 0 1 1 0 0 0
%e 1 0 0 0 0 1 1 1 1 0
%e 1 0 1 0 1 0 1 1 0 0
%e 1 0 1 1 0 0 1 1 1 1
%e 1 1 1 1 0 1 0 1 1 0
%e 0 1 1 0 1 1 1 1 1 0
%e 1 0 1 1 1 1 0 1 0 1
%e a(11) >= 23, from _Dmitry Kamenetsky_, Jan 05 2018:
%e 0 1 1 0 0 0 1 0 1 1 1
%e 0 0 0 0 1 0 0 0 0 0 1
%e 0 0 1 1 0 0 1 0 0 0 1
%e 0 1 1 0 1 0 1 1 1 0 0
%e 0 0 1 0 1 1 0 0 1 1 1
%e 1 1 0 1 1 0 0 0 1 1 0
%e 1 0 0 1 1 0 0 1 0 1 1
%e 1 0 0 1 0 1 1 0 1 1 0
%e 1 1 1 1 1 1 0 0 0 0 0
%e 1 0 0 1 1 1 1 1 1 0 1
%e 1 0 1 0 0 0 0 0 0 0 0
%e 1 0 0 0 0 1 0 1 1 1 1
%e 0 1 0 0 1 0 1 0 0 1 0
%e 1 1 0 0 0 0 0 0 1 0 1
%e 1 1 1 0 0 1 1 1 0 1 1
%e 1 1 0 0 0 1 1 0 0 0 0
%e 0 0 0 1 0 0 1 1 0 1 0
%e 1 0 1 1 1 1 1 0 0 1 1
%e 1 1 1 1 0 0 0 1 0 0 1
%e 0 1 1 1 0 0 0 1 1 1 0
%e 0 1 1 0 1 1 0 1 0 1 0
%e 0 0 0 0 0 1 0 1 0 0 0
%e 0 1 0 1 0 1 0 0 0 0 1
%Y Cf. A289972.
%K nonn,more,nice
%O 0,2
%A _David Bevan_, Jan 06 2004
%E Edited by _N. J. A. Sloane_, Mar 29 2016 and Apr 08 2016
%E a(10) from _Fausto A. C. Cariboni_, Jul 17 2017
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