Summary: Large sets of nearly orthogonal vectors
It is shown that there is an absolute positive constant > 0, so that for all positive integers k
and d, there are sets of at least d log2(k+2)/ log2 log2(k+2)
nonzero vectors in Rd
, in which any k + 1
members contain an orthogonal pair. This settles a problem of F¨uredi and Stanley.
For two positive integers d and k, let (d, k) denote the maximum possible cardinality of a set of
nonzero vectors in Rd such that among any k + 1 members of the set there is an orthogonal pair.
More generally, for three positive integers d and k l 1, let (d, k, l) denote the maximum possible
cardinality of a set P of nonzero vectors in Rd such that any subset of k + 1 members of P contains
some l+1 pairwise orthogonal vectors. Thus (d, k) = (d, k, 1). Trivially, (d, 1) = d and Rosenfeld
 proved, using an interesting algebraic argument, that (d, 2) = 2d for every d. F¨uredi and Stanley
 observed that (2, k) = 2k, and proved that (4, 5) 24, that for every fixed d and l the limit
limk(d, k, l)/k is equal to its supremum, and that for every fixed l there exists some l > 0 and
d0 such that this supremum is at least (1 + l)d for all d > d0 and at most
(1 + o(1)) d/(2l)((l + 1)/l)d/2-1