An easy proof of Gowers’ FINk theorem
Ryszard Frankiewicza
Sławomir Szczepaniakb
a Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa,
Poland, e-mail: [email protected], Tel. No. +48 512 899 576
b Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa,
Poland, e-mail: [email protected], Tel. No. +48 668 108 732
ARTICLE INFO
ABSTRACT
Keywords
Gowers Theorem
Hindman Theorem
Ramsey Theory
A new proof of Pigeon Principle of Gowers is found.
The proof does not use of the concept of ultrafilter.
The purpose of the paper is to give an elementary proof of the theorem of
Gowers ([6]) being a generalization of Hindman Theorem ([7]). The presented
proof is purely combinatorial and (on the contrary to original Gowers’ proof)
do not use the theory of ultrafilters as well as the full strength of the Axiom
of Choice. Our method mimics Baumgartner’s proof of Hindman Theorem
([3]). In fact our proof stands in the same relation to Baumgartner’s as the
ultrafilter proof of Hindman Theorem to the proof of Gowers Theorem; we
refer the reader to [5] for comparison. In particular the main ingredient, Finite Gowers’ FINk Theorem, replaces Gowers’ Lemma 3 from [6] (known
also as the lemma on stabilization of continuous endomorphisms, cf. [2]).
The demand for combinatorial was raised for example in [10]: in hindsight one can see that Baumgartner’s proof of Hindman’s theorem also uses a
combinatorial forcing and this suggests the natural question of whether there
is an analogous proof that would establish Gowers’s pigeon hole principle for
FINk , originally proved using the methods of topological dynamics. The paper is organized in three sections. The first one consists of notations and
basic facts. In the second section we prove Gowers’ FINk Theorem.
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1. Notations and simple observations.
Throughout the paper we use letters i, j, k, l, m, n for nonnegative natural
numbers and by ω we denote their universe. Let also N := ω \ {0}. We prefer
to treat numbers as ordinals; thus for example k + 1 = {0, 1, . . . , k} and i < 2
means i ∈ {0, 1}. Gowers’ Pigeon Principle also known as Gowers Theorem
is a Ramsey-type theorem about particular families of functions (sequences)
and that is why we need specific notations; some of them we borrow from
[10]. We put supp(p) = {n < ω : p(n) 6= 0} for the support of a function
p : ω → k + 1, k < ω, while by rng(p) we we denote its range p[ω]. Define
the main object in the paper by
FINk = {p ∈ω (k + 1) : |supp(p)| < ω & k ∈ rng(p)} .
Thus FIN0 is a singleton of the null function FIN0 = {{(i, 0) : i < ω}} = {O}.
We equip the collections FINk , k < ω, with an ordering defined as
p<q
if
max supp(p) < min supp(q),
p, q ∈ FINk .
Moreover, for functions comparable under this ordering we define their sum
as pointwise sum of functions. It makes the structure (FINk , +) a partial
semigroup (cf. [10]). Regarding sums we shall use the following conventions.
If we write p + q we always implicite assume that p · q = O (pointwise
multiplication), i.e. supp(p) ∩ supp(q) = ∅.
We say that a family B ⊆ FINk is a block sequence if it can be enumerated
by natual numbers increasingly with respect to the ordering <. Note that
there exists exactly one such enumeration, therefore we always assume that
any block sequence B comes with its increasing enumeration (bn ), where, unless otherwise stated, we use the convention that elements of a block sequence
C are denoted by small letters c.
For k ∈ N define the following tetris operation T : FINk → FINk−1 by
T (p)(n) = max{p(n)−1, 0}, n < ω, and by T n denote its nth iteration, n 6 k;
of course T 0 = id. The tetris operation is a surjective and additive function
which means that T (p + q) = T (p) + T (q) for p, q ∈ FINk .
Any block sequence B generates a set called combinatorial space hBi
defined as the smallest subfamily of FINk consisting of B and closed on the
summing and tetris operations:
(
)
X
hBi :=
T k−f (n) (bn ) : f ∈ FINk .
n<ω
Note that a combinatorial space is a well-defined subfamily of FINk and FINk
is itself a combinatorial space generated by {{(i, kδin ) : i < ω} : n < ω} (here
2
δ denotes Kronecker delta). On the collection of all block sequences of FINk
we consider the following partial ordering
B0 4 B1
iff B0 ⊆ hB1 i.
We say that a family F ⊆ FINk is B-large if for all B 0 4 B the family F
meets the combinatorial space generated by B 0 .
Observe the following facts regarding large families. The first fact means
that ’somewhere’ B-large (C-large for some C 4 B) families form a coideal
and the second gives that this coideal is nonpricipal (cf. [10] for this notions).
Fact 1 If B-large family F ⊆ FINk is written as a finite sum of its subfamilies,
then one of them is B 0 -large for some B 0 4 B.
Indeed, if F = F0 ∪ F1 and F0 is not B-large then F ∩ hB 0 i = ∅ for some
B 0 4 B. Then by B-largeness of F for any B 00 4 B 0 4 B it holds ∅ =
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F ∩ hB 00 i = F1 ∩ hB 00 i (cf. [5]).
Fact 2 If F is B-large then so is F/s := {p ∈ F : s < p} for any s ∈ FINk .
Indeed, if for some s ∈ FINk the family F/s is not B-large then we would get
hCi ∩ F/s = ∅ for some block sequence C 4 B. Then however hC/si ∩ F = ∅
and thus a block sequence C/s 4 B would witness that F is not B-large.
In the forthcoming sections we shall give a proof of
Gowers’ FINk Theorem Let k ∈ N. For any finite coloring of FINk there
exits monochromatic combinatorial subspace of FINk .
Observe that by the simple induction we can restricts ourselves to 2-coloring
of combinatorial spaces. Furthermore by the discussion on large families any
2-coloring produces monochromatic B-large family. Thus we need only to
prove the following
Gowers’ FINk Theorem If F is a B-large family in FINk , k ∈ N, then
there is a block sequence B with hCi ⊆ F.
2. Proof of Gowers’ FINk Theorem.
We begin with the finite version of Gowers’ FINk Theorem. First, denote
by `n∞ the Banach space Rn with the supremum norm || · ||∞ and for Banach
space X ⊆ `∞ := {x ∈ Rω : ||x||∞ < ∞} put SX := {x ∈ X : ||x|| = 1}
+
and SX
:= SX ∩ [0, ∞)ω for the unit sphere of X and its positive hemisphere,
respectively. For other undefined notions from Banach Space Theory we
refer to [1]. We deduce the finite version of Gowers’ FINk Theorem from the
following result of V.Milman which is Theorem 6 from [8] in the language of
[9] (Theorem 0.0.4 and Theorem 1.1.18 (1)-(8)). We stress that this result
does not use uncountable versions of Axiom of Choice.
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Milman Theorem Given ε > 0, n ∈ N there exists a natural number
is divided into disjoint sets A0 , A1 then there
M = M (ε, n) so that if S`M
∞
n
exists i < 2 and a block subspace E of `M
∞ isometric to `∞ such that
SE ⊆ (Ai )ε := {x ∈ S`M
: ||x − a|| < ε for some a ∈ Ai }.
∞
Finite Gowers’ FINk Theorem If n ∈ N and F ⊆ FINk is B-large then
hF i ⊆ F for some n-element family F of k-tuples from FINk .
Proof: Let n ∈ N and ε > 0 be such that (1+ε)−(k−1) < ε. Put M = M (ε, n)
given by Milman Theorem. Consider now the following isomorphic copy
∆k of FINk (cf. [6]): g ∈ ∆k iff g is finitely supported function such that
1 ∈ rng(g) ⊆ {(1 + ε)−i : i < k}. It is evidently a net in S`+M isomorphic to
∞
FINk via the following bijection Φ : ∆k 7→ FINk
log x(m)
,0 .
Φ(x)(m) := max k −
log(1 + ε)−1
Here being isomorphic means preserving block sequences in the sense of Theory of Banach Spaces and in the sense described in the first section. Now
take a block subsequence E of standard basis of `M
∞ which generates a subspace isometric to `n∞ inside (Φ−1 [F])ε = Φ−1 [F]. Then it is immediate that
F := Φ[E] do the work.
As it was mentioned the following proof is mostly (modulo notation and
new notions) due to Baumgartner, only the first lemma being different (cf.
Theorem 6.5 from [4]). In this section we shall use hBi∗ for hBi \ B and
hBi(2) for block sequences from hBi of length 2.
Lemma 3 Let F be B-large. There exist s ∈ FINk and a block sequence
B 0 4 B such that the following family {p ∈ F/s : h{s, p}i∗ ⊆ F} is B 0 -large.
Proof: Towards a contradiction assume that for all pairs (s, D) of functions
s ∈ FINk and block sequences D 4 B there is a block sequence E 4 D such
that h{s, p}i∗ * F for all p ∈ hEi ∩ F/s. Denote this assumption by (∗).
Starting with B−1 := B and arbitrary s0 ∈ hB−1 i we define recursively a
block sequence S = (sn ) 4 B along with 4-decreasing sequence Bn of block
sequences such that the following condition (∗∗) is satisfied. For all n < ω,
s ∈ h{si : i 6 n}i and p ∈ hBn i ∩ F/sn it holds h{s, p}i∗ * F. In order to see
that such a recursion is possible assume that we have already constructed a
partial sequence {(si , Bi ) : i 6 n} satisfying condition (∗∗). Define arbitrarily
sn+1 ∈ hBn i and enumerate h{si : i 6 n + 1}i := {tl : l 6 N }, N < ω.
Starting with Bn−1 := Bn define 4-decreasing sequence {Bnl : l 6 N } of block
sequences such that h{tl , p}i∗ * F for all p ∈ hBnl i ∩ F/s. The choice of
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Bnl is guaranteed by (∗). By putting Bn+1 := BnN one can easily see that a
partial sequence {(si , Bi ) : i 6 n + 1} possesses required properties.
Observe now that by the construction we obtain h{s0 , s1 }i∗ * F or s1 ∈
/F
(2)
for any (s0 , s1 ) ∈ hSi . In particular h{s0 , s1 }i * F for any (s0 , s1 ) ∈ hSi(2) .
This contradicts Finite Gowers’ FINk Theorem and finishes the proof.
Lemma 4 Let F be B-large. There exist s ∈ F and a block sequence B 0 4 B
such that the following family {p ∈ F/s : h{s, p}i∗ ⊆ F} is B 0 -large.
Proof: Starting with B−1 := B, F−1 := F we define recursively a block
sequence S = (sn ) 4 B along with ⊆-decreasing sequence of families (Fn )
and 4-decreasing sequence Bn of block sequences such that the following
conditions (∗)n are satisfied for all n < ω
• sn ∈ hBn−1 i and sn−1 < sn ,
• Fn := {p ∈ Fn−1 /sn : h{sn , p}i∗ ⊆ Fn−1 } is Bn -large.
In order to see that such a recursion is possible assume that we have already
constructed a partial sequence {(si , Bi , Fi ) : i 6 n} satisfying condition (∗)n .
Since the family Fn is Bn -large it is also Bn /sn -large. Therefore we can apply
Lemma 3 to F := Fn and B := Bn /sn and find sn+1 ∈ hBn /sn i and a block
sequence Bn+1 4 Bn /sn such that Fn+1 := {p ∈ Fn /sn+1 : h{sn+1 , p}i∗ ⊆ Fn }
is Bn+1 -large. Note that sn+1 > sn as sn+1 ∈ hBn /sn i. This finishes the definition of {(si , Bi , Fi ) : i 6 n + 1} which clearly satisfies condition (∗)n+1 .
Using backward induction one can observe that for all m 6 n < ω it holds
Fn ⊆ {p ∈ Fn−m−1 /sn : h{si : n − m 6 i 6 n} ∪ {p}i∗ ⊆ Fn−m−1 }
and in particular Fen = {p ∈ F/sn : h{si : i 6 n} ∪ {p}i∗ ⊆ F} ⊇ Fn
(∗∗).
Considering S 4 B by B-largeness of F there is a function s ∈ F ∩ hSi. In
particular there is M < ω such that s ∈ h{si : i 6 M + 1}i. Furthermore,
if h{si : i 6 M + 1} ∪ {p}i∗ ⊆ FM then also h{s, p}i∗ ⊆ FM , p ∈ FM /sM +1 .
Hence FM ⊆ FeM ⊆ {p ∈ FM /sM +1 : h{s, p}i∗ ⊆ FM } ⊆ {p ∈ F/s :
h{s, p}i∗ ⊆ F}. Thus the lemma holds with B 0 := BM as FM was BM -large
and contained in {p ∈ F/s : h{s, p}i ⊆ F} by the above inclusions.
Proof Gowers’ FINk Theorem:
Starting with B−1 := B, F−1 := F we define recursively a block sequence
S = (sn ) 4 B along with ⊆-decreasing sequence of families (Fn ) and 4decreasing sequence Bn of block sequences such that the following conditions
are satisfied for all n < ω
• sn ∈ Fn−1 ∩ hBn−1 i and sn−1 < sn ,
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• Fn := {p ∈ Fn−1 /sn : h{sn , p}i∗ ⊆ Fn−1 } is Bn -large.
The construction of the above objects is the same as in Lemma 3.3 with
the one exception. To find sn+1 we use Lemma 4 instead of Lemma 3.
Let us check that such a S 4 B works. Take an arbitrary s ∈ hSi and
let M < ω be such that s ∈ h{si : i 6 M + 1}i. Since sM +1 ∈ FM by
g
the inclusion (∗∗) from Lemma 3 we obtain sM +1 ∈ F
M . In particular
h{si : i 6 M } ∪ {sM +1 }i∗ ⊆ F and s ∈ F. By arbitrariness of s ∈ hSi the
theorem follows.
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