On Higman’s lemma
Josef Berger
Ernst Moritz Arndt Universität Greifswald
Proof
Bern
September 2013
Embedding of words
010 v 001011
0101110101 v 0101110101
0011 v 01010101
010101 v 100011111010111
Embedding of words
We define a binary relation on {0, 1}∗ by
() v ()
uvw
u v w0
uvw
u v w1
uvw
u0 v w 0
uvw
u1 v w 1
Higman’s lemma
Fix a sequence (un ) of binary words.
HL there are i, j such that
I i <j
I ui v uj
Dickson’s lemma
Fix f : N → Nn .
DLn there are i, j such that
I i <j
I f (i) ≤ f (j) (component-wise)
Observation
∀n (DLn )
“←−” is easy
“−→” is not easy
←→
HL
Conventions
For Dickson, work with
f : N → Nn+ .
For Higman, work with
g : N → {0, 1}∗0 ,
where
{0, 1}∗0 = {0u | u ∈ {0, 1}∗ } .
HL −→ DLn
For k = (k1 , . . . , kn ) ∈ Nn+ set
. . 1} 0 . . . .
κ(k) = |0 .{z
. . 0} |1 .{z
k1
k2
Fix a sequence ki in Nn+ . By HL, there are i < j with
κ(ki ) v κ(kj ),
this implies
ki ≤ kj .
The weight notion
Assign a weight Φ(u) to u as indicated by
u = |{z}
000 11111
00 |{z}
111
| {z } |{z}
one
two
three four
Φ(u) = 4.
The weight lemma
Lemma
(2 · |u| ≤ Φ(w )) → u v w
Example: |u| = 3, Φ(w ) = 7
u1 u2 u3 v |0 . . . 01
{z . . . 1} |0 . . . 01
{z . . . 1} |0 . . . 01
{z . . . 1} 0 . . . 0
embed u1 here embed u2 here embed u3 here
Forcing a certain weight upon u ∈ {0, 1}∗0
For fixed n assign u!n of weight n to any u.
If Φ(u) = n, we define u!n = u.
If Φ(u) < n, let u!n be the shortest extension of u which has
weight n.
0111000!6 = 0111000101
If Φ(u) > n, let u!n be the longest restriction of u which has
weight n.
011100011100000001!3 = 0111000
The second weight lemma
Lemma
For u, w ∈ {0, 1}∗0 and n ∈ N with
Φ(u) ≤ Φ(w ) ≤ n
we have
u!n v w !n → u v w
The second weight lemma
The second weight lemma follows from
ui v vi → u v v
and
ui v v → u v v .
example: n = 6,
000110001111
|
{z
} 01 v 000111111000011111110
|
{z
}1
u with weight 4
w with weight 5
Remember:
Instead of k = (k1 , . . . , kn ) ∈ Nn+ , consider
κ(k) = |0 .{z
. . 0} |1 .{z
. . 1} 0 . . . .
k1
k2
A natural correspondence
Let Wn denote the elements of {0, 1}∗0 with weight n.
There is an order-preserving bijection
κn : Nn+ → Wn .
DLn −→ HL
Fix a sequence (uk )k∈N in {0, 1}∗0 . Set n = 2 · |u0 |.
DLn+1 yields a pair i < j with
−1
(κ−1
n (ui !n ), Φ(ui )) ≤ (κn (uj !n ), Φ(uj )).
Φ(uj ) > n
u0 v uj by the weight lemma
Φ(uj ) ≤ n Φ(ui ) ≤ Φ(uj ) and ui !n v uj !n yield ui v uj (by the
second weight lemma)