Another formulation is just to say that your "directions" are points on the standard projective plane. Points in the projective plane are lines through the origin in 3-space, so this automatically gives you the equivalence you want under rescaling and negation. I am pretty sure that f(6) = 2. Say ABC is a triple. Two triples cannot share more than one point. If no triples share a point, then the best we can do is ABC+DEF. If two triples *do* share a point, w.l.o.g. ABC+CDE, there is no way that F can complete a third triple. It would have to group with one of each of the existing triples, w.l.o.g. ADF. But then A is orthogonal to D, so ACD, which violates the single-overlap lemma. I forget the axioms for a Steiner triple system, but this feels very familiar. On Thu, Jan 30, 2014 at 6:24 PM, Warren D Smith <warren.wds@gmail.com>wrote:

A "direction" is a 3-vector (x,y,z) obeying either x>0, or x=0 and y>0, or x=y=0 and z>0, and x^2+y^2+z^2=1.

The point of that is (a) to forbid 0,0,0 and (b) I want to regard negative directions and rescaled directions as same thing as original directions, so only allow one.

Let f(N) be the maximum possible number of (unordered) triples of mutually-orthogonal directions in a set of N distinct directions. What can be said about f(N)?

Here is a table of lower bounds L and upper bounds U on f(N) for small N. Which probably are both pretty weak. N=3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 L=1 1 2 2 3 3 4 5 5 6 6 7 8 9 9 10 11 11 12 13 14 14 15 15 16 17 18 18 19 19 U=1 1 2 4 7 8 12 13 17 20 26 28 35 37 44 48 57 60 70 73 83 88 100 104 117 121

-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun