Graphs and Combinatorics (1992) 8:391 394
Graphs and
Combinatorics
Research Problems
9 Springer-Verlag1992
In this column Graphs and Combinatorics publishes current research problems whose proposer
believes them to be within reach of existing methods.
Manuscripts should preferably contain the background of the problem and all references
known to the author. The length of the manuscript should not exceed two type-written pages.
Manuscripts should be sent to:
P. Frankl or J. Akiyama
Graphs and Combinatorics
Department of Mathematics
Tokai University
Hiratsuka, 259-12
Japan
Sets of Vectors with Many Orthogonal Pairs:
Abstract. What is the most number of vectors in
R d such that any k + 1 contain an orthogonal
pair? The 24 positive roots of the root system F 4 in R 4 show that this number could exceed dk.
1. Almost Orthogonal Systems
T h e p u r p o s e of this p a p e r is to call a t t e n t i o n to the following p r o b l e m , whose
s o l u t i o n m i g h t involve different b r a n c h e s of c o m b i n a t o r i c s . Let ~ be a collection
of n o n z e r o vectors of the d - d i m e n s i o n a l euclidean space. D e n o t e by a ( ~ ) the size
of the largest subset A c ~ such that no two of the vectors v 1, v 2 e A are o r t h o g o n a l .
Finally, o~td)(k)s t a n d s for m a x { l ~ ] : ~ c Rd, 0~(~) _~ k}. T h a t is, ~(d)(k) is the maxim u m n u m b e r of vectors such t h a t a n y k + 1 of t h e m c o n t a i n an o r t h o g o n a l pair.
C o n s i d e r i n g k o r t h o g o n a l basis we have
c~(a)(k) _> dk
O n the o t h e r h a n d , e(d)(k) is finite, e(a)(k) _< R(d + 1, k + 1) _<
(1)
d
, where R is
the usual Ramsey number [3]. T h e p r o b l e m of d e t e r m i n i n g c~(a)(k)is due, of course,
to E r d 6 s [5]. H e a s k e d w h e t h e r e q u a l i t y holds in Eq. 1.
* The research was done while the author Z. Fiiredi visited the Department of Mathematics at
Massachusetts Institute of Technology. Research supported in part by the Hungarian National
Science Foundation under grant No. 1812. Research supported in part by NSF grants.
392
Research Problems
Obviously, ~d)(1) = d, and Rosenfeld [5] gave an ingenious algebraic p r o o f for
~(d)(2) = 2d. The p r o o f of ~(2)(k) = 2k is easy (see Sect. 4).
2. The Root System F4
The following example shows that
~4(5) _> 24 > 2 0 ,
(2)
i.e., we do not always have equality in Eq. 1. Let ea, e2, e3, e 4 be the four unit vectors,
and let ~ consist of the following 24 vectors
:= {ea,...,e4} U {e i + ej: i < j } U {e 1 _+ e2 _+ e3 + e4}.
We claim that any 6-element subset of N contains an o r t h o g o n a l pair. Suppose, on
the contrary, that A ~ ~ has no o r t h o g o n a l pair and [A] = 6. Split ~ into 6 bases:
1000
0100
0010
0001
1
1
0
0
0
0
1
--1
0 1
1
0 1 -1
0
0
1
1
1
--1
0
0
0
0
1
1
1 --1
1
-1
1
1
1
1
-1
-1
0
0
0
0
1
-1
1
t
1
0
1 0 --I
0 1
0
0 1
0
1
0
1
--1
-1
-I
1
1
0
1
1
1
1
--1
1
1
--1
-1
1
--1
1
-1
-I
1
1
We obtain that A meets each basis in exactly one element. Observe, that the vector
v e ~ has the same role as - v and, similarly, all i'th coordinates can be replaced
by their opposite ones (i.e., x by - x ) keeping all the orthogonalities. In this way,
the first coordinate has no exceptional role, so we m a y suppose that el ~ A. Simultaneously, we m a y suppose that (1, 1, 1, 1) E A, too. Then, from the second, third and
fourth basis the (non-orthogonal) vector of A is (i, 1, 0, 0), (1, 0, 1, 0) and (1, 0, 0, 1),
respectively. However, each vector of the sixth basis is o r t h o g o n a l to one of el + %
a contradiction.
[]
3. Asymptotic Results
The next inequalities imply that limk-,~ o:(d)(k)/k equals to its s u p r e m u m , ~(d)
:dd)(k) + :t(d)(l) _< odd)(k + l)
(3a)
~(d)(k) + ~(S)(k) _< ~(a+Y)(k)
(3b)
We get from Eq. 2
~(4~(k) _> 24Lk/5J + cd4)(k - 5 Lk/5J) > 4.8k - 4.
Research Problems
393
Let 5e be an S(d, 4, 2) Steiner system (d - 1 or 4 (mod 12), d > 13), and define
c R d as follows. Consider the d unit vectors, the vectors of form el _+ ej
(1 < i < j < d), and finally the vectors of the form ei + e~_+ eu__ ev(1 < i < j < u <
v<d,{i,j,u,v}~5~). We have that [ ~ [ = l + ( 5 / 3 ) d ( d - 1 ) , and c~(~)=
1 + (4/3)(d - 1). (In case oflA] > 16, ifA is an orthogonal-free subset of~', then the
supports of the vectors from A have a common element, etc.) This implies
~d) > 4 d _ 3.
Even more, Eqs. 2 and 3 imply that there is no equality in Eq. 1 except, eventually,
if either d < 3, or k < 4.
Blokhuis showed us that the following 8-dimensional example gives ~ts)(8) >
120. Then, using the same procedure as above, for large k or d, one gets better ratios.
: = {ei +__ej: 1 _< i < j _< 8} U {e I + e 2 _+"" _+ e8}.
4. A Geometric Upper Bound
Let Sd-1 denote the unit sphere in R a, and let C a-~ := {x s Sd-l" I(x, el) I > 1/~2}.
This set consists of two spherical caps. Two vectors from the same cap determine
an acute angle, and from different caps form an obtuse angle. An averaging argument shows that ~r
> Area(C~-~)/Area(S~-l)I~l. This implies ~t2)(k) < 2k, and
cd3)(k) ___(2 + x/~)k ~ 3.41... k.
In general we obtain
~ta)(k)/k < (1 + o(1)) x/~d 2a/2
(4)
Remark. Witsenhausen [1] proposed the following problem. What is p(d), the
largest area of a measurable subset C c Sd-1 such that no vectors of C are
orthogonal? Even the exact value of #(3) is unknown. The best known lower bound
(and the conjectured exact value) is given by Cd-~. The best upper bound
(#(d) < (1 + o(1))1.13 -a) is due to Frankl [1], and follows from Eq. 5 (see later).
Obviously, the essence of Eq. 4 is the upper bound Ot(d)(k) <~(Area(S~-~)/#(d))k.
Considering, e.g., the first orthant of R d and its opposite (instead of Ca-a), we
obtain the simpler but weaker ~d)(k) < 2a-lk, which holds for all d and k.
5. Further Examples from _+1 Codes
Let Qa be the set of vectors of R a of the form (el . . . . . ca) where e~is 1 or - 1. We have
IQd] = U. Larman and Rogers [4] raised the question in 1972 that what is the
maximum number re(d) of elements of Qd so that no two are orthogonal, i.e., the
determination of ~(Qd). The really interesting case is when the dimension is divisible
by 4. Frankl and R6dl [2] proved that m(4q) < (2 - c)4q for some small but positive
c. Frankl [ 1] proved the following exact statement: If q is an odd primepower, then
q-~ ( 4 q m(4q) = 4 ~
i=0
i
1)
< 44q/33q =
(1.754...)4q
(5)
394
Research Problems
This and Eq. 3 imply the exponential lower b o u n d (27/16) q < ~(4q), and in general,
for d > d o
1.13 a < c~~a).
Frankl uses Eq. 5 to estimate the chromatic number z(d) of R a, i.e., the chromatic
n u m b e r of the graph with vertex set R a and two points are joined if their Euclidean
distance is 1. Even the value of :~(2) is unknown.
6. Sets with M o r e Orthogonal Pairs
Denote by ~n)(k) the m a x i m u m n u m b e r of vectors such that any k + 1 of them
contains l + t pairwise o r t h o g o n a l ones. Thus ~]d)(k)= ~(a)(k). All the above
questions and examples can be generalized. F o r example, considering a larger cone
(l(x, el)l > 1//,,/~+ 1) instead o f E q . 4 we get
~f~d(l + 1~ (a/2)-1
a~a)(k) _< (1 +o(1))X/ 2 / \
l
J
k.
Frankl and R6dl [2] proved that there exists a positive c = c(l) such that every
subset of Q4n of size m o r e than (2 - c)*" contains 1 + 1 pairwise o r t h o g o n a l vectors.
This implies that ~a)(k)/k is exponentially large in d.
We conjecture that there is a constant g = a(1) such that ~la)(k) < (dk) g.
References
1. Frankl, P.: Orthogonal vectors in the n-dimensional cube and codes with missing distances,
Combinatorica 6, 279-285 (1986)
2. Frankl, PI, RiSdl, V.: Forbidden intersections, Trans. Amer. Math. Soc. 300, 259-286 (1987)
3. Graham, R.L., Rothschild, B.L., Spencer, J.: Ramsey Theory, Wiley Interscience, 1980. (2nd
edition 1990)
4. Larman, D.G., Rogers, C.A.: The realization of distances whithin sets in eucledian space,
Mathematika 19, 1-24 (1972)
5. Moshe Rosenfeld, Almost orthogonal lines, in E d, DIMACS Series in Discrete Math. 4, 489-492
(1991)
Zolthn Fiiredi I and Richard Stanley z
Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, Budapest 1364,
Hungary e-mail: hl 154fur@ella.hu, and Department of Mathematics University of Illinois,
Urbana IL 61801, USA zoltan@symcom.math.uiuc, edu
z Department of Mathematics Massachusetts Institute of Technology, Cambridge, MA
02139-4307, USA e-mail: rstan@math.mit.edu
Received: August 21, 1991