OFFSET
0,2
COMMENTS
Define addition on {0,1}^n componentwise (ordinary addition, not addition modulo 2, so the result lies in {0,1,2}^n, not necessarily {0,1}^n). We say a subset of {0,1}^n is Sidon iff the only solutions to a+b = c+d, with a,b,c,d in the set, are the trivial ones: a=c, b=d or a=d, b=c.
a(7) >= 23, a(8) >= 32, a(9) >= 45, a(10) >= 63, a(11)>=87, a(12)>=120, a(13)>=169, a(14)>=237, a(15) >= 334, a(16) >= 472, a(17) >= 662, a(18) >= 864.
Conjecture: a(n) is asymptotic to 2^(n/2+1).
LINKS
G. Cohen, S. Litsyn and G. Zémor, Binary B_2-Sequences: A New Upper Bound, Journal of Combinatorial Theory, Series A 94 (2001): 152-155.
B. Lindström, On B_2-sequences of vectors, Journal of Number Theory 4 (1972): 261-265.
EXAMPLE
When n=2, the elements of {0,1}^n are (0,0),(0,1),(1,0),(1,1). The entirety of {0,1}^2 is not Sidon, because (0,0)+(1,1) = (0,1)+(1,0). But this is the only nontrivial solution to a+b = c+d in {0,1}^2, so removing any one element results in a Sidon set, so a(2)=3.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Asier Calbet Rípodas, Aug 02 2019
STATUS
approved