proofs

Communicating without errors 247
On the other hand, for x = ( 1 m , . . ., 1 m
) we obtain
µ T
(G) ≤ µ(x) = | 1 m (v(1) + . . . + v (m) )| 2 = |u| 2 = σ T
,
and so we have proved
µ T
(G) = σ T
. (6)
In summary, we have established the inequality
α(G) ≤ 1
σ T
(7)
for any orthonormal respresentation T with constant σ T
.
(C) To extend this inequality to Θ(G), we proceed as before. Consider
again the product G × H of two graphs. Let G and H have orthonormal
representations R and S in R r and R s , respectively, with constants σ R
and σ S
. Let v = (v 1 , . . .,v r ) be a vector in R and w = (w 1 , . . . , w s ) be
a vector in S. To the vertex in G × H corresponding to the pair (v, w) we
associate the vector
vw T := (v 1 w 1 , . . .,v 1 w s , v 2 w 1 , . . .,v 2 w s , . . .,v r w 1 , . . .,v r w s ) ∈ R rs .
It is immediately checked that R × S := {vw T : v ∈ R, w ∈ S} is an
orthonormal representation of G × H with constant σ R
σ S
. Hence by (6)
we obtain
µ R×S
(G × H) = µ R
(G)µ S
(H).
For G n = G × . . . × G and the representation T with constant σ T
this
means
µ T
n(G n ) = µ T
(G) n = σ n T
and by (7) we obtain
α(G n ) ≤ σ −n
T , n √ α(G n ) ≤ σ −1
T .
Taking all things together we have thus completed Lovász’ argument:
Theorem. Whenever T = {v (1) , . . .,v (m) } is an orthonormal
representation of G with constant σ T
, then
Θ(G) ≤ 1
σ T
. (8)
Looking at the Lovász umbrella, we have u = (0, 0, h= 1 4 √ 5 )T and hence
σ = 〈v (i) , u〉 = h 2 = 1 √
5
, which yields Θ(C 5 ) ≤ √ 5. Thus Shannon’s
problem is solved.
“Umbrellas with five ribs”