proofs

100 Borsuk’s conjecture
To obtain a general bound for large d, we use monotonicity and unimodality
of the binomial coefficients and the estimates n! > e( n e )n and n! < en( n e )n
(see the appendix to Chapter 2) and derive
∑q−2
( ) ( )
4q − 3 4q
< q
i q
i=0
Thus we conclude
From this, with
f(d) ≥ g(q) =
= q (4q)!
q!(3q)! < q
2 4q−4
q−2 ∑ ( 4q−3
i
i=0
e 4q ( 4q
e ( q
e
e
) 4q
) q
e
( 3q
e
) 3q
= 4q2
e
) > e ( 27
) q.
64q 2 16
( 256
) q.
27
d = (2q − 1)(4q − 3) = 5q 2 + (q − 3)(3q − 1) ≥ 5q 2 for q ≥ 3,
√ √
q = 5 8 + d
8 + 1 64 > d
8 , and ( 27
) √ 1
8
16
> 1.2032,
we get
f(d) >
e
√ √
13d (1.2032) d
> (1.2) d
for all large enough d. □
A counterexample of dimension 560 is obtained by noting that for q = 9 the
quotient g(q) ≈ 758 is much larger than the dimension d(q) = 561. Thus
one gets a counterexample for d = 560 by taking only the “three fourths”
of the points in S that satisfy x 21 + x 31 + x 32 = −1.
Borsuk’s conjecture is known to be true for d ≤ 3, but it has not been
verified for any larger dimension. In contrast to this, it is true up to d = 8
if we restrict ourselves to subsets S ⊆ {1, −1} d , as constructed above
(see [8]). In either case it is quite possible that counterexamples can be
found in reasonably small dimensions.
References
[1] K. BORSUK: Drei Sätze über die n-dimensionale euklidische Sphäre, Fundamenta
Math. 20 (1933), 177-190.
[2] A. HINRICHS & C. RICHTER: New sets with large Borsuk numbers, Discrete
Math. 270 (2003), 137-147.
[3] J. KAHN & G. KALAI: A counterexample to Borsuk’s conjecture, Bulletin
Amer. Math. Soc. 29 (1993), 60-62.
[4] A. NILLI: On Borsuk’s problem, in: “Jerusalem Combinatorics ’93” (H.
Barcelo and G. Kalai, eds.), Contemporary Mathematics 178, Amer. Math.
Soc. 1994, 209-210.
[5] A. M. RAIGORODSKII: On the dimension in Borsuk’s problem, Russian Math.
Surveys (6) 52 (1997), 1324-1325.
[6] O. SCHRAMM: Illuminating sets of constant width, Mathematika 35 (1988),
180-199.
[7] B. WEISSBACH: Sets with large Borsuk number, Beiträge zur Algebra und
Geometrie/Contributions to Algebra and Geometry 41 (2000), 417-423.
[8] G. M. ZIEGLER: Coloring Hamming graphs, optimal binary codes, and the
0/1-Borsuk problem in low dimensions, Lecture Notes in Computer Science
2122, Springer-Verlag 2001, 164-175.