In mathematics, Gowers' theorem, also known as Gowers' Ramsey theorem and Gowers' FINk theorem, is a theorem in Ramsey theory and combinatorics. It is a Ramsey-theoretic result about functions with finite support. Timothy Gowers originally proved the result in 1992, motivated by a problem regarding Banach spaces. The result was subsequently generalised by Bartošová, Kwiatkowska, and Lupini.
Definitions The presentation and notation is taken from Todorčević, and is different to that originally given by Gowers. For a function f : N → N {\displaystyle f\colon \mathbb {N} \to \mathbb {N} } , the support of f {\displaystyle f} is defined supp ( f ) = { n : f ( n ) ≠ 0 } {\displaystyle \operatorname {supp} (f)=\{n:f(n)\neq 0\}} . Given k ∈ N {\displaystyle k\in \mathbb {N} } , let F I N k {\displaystyle \mathrm {FIN} _{k}} be the set
F I N k = { f : N → N ∣ supp ( f ) is finite and max ( range ( f ) ) = k } {\displaystyle \mathrm {FIN} _{k}={\big \{}f\colon \mathbb {N} \to \mathbb {N} \mid \operatorname {supp} (f){\text{ is finite and }}\max(\operatorname {range} (f))=k{\big \}}}
If f ∈ F I N n {\displaystyle f\in \mathrm {FIN} _{n}} , g ∈ F I N m {\displaystyle g\in \mathrm {FIN} _{m}} have disjoint supports, we define f + g ∈ F I N k {\displaystyle f+g\in \mathrm {FIN} _{k}} to be their pointwise sum, where k = max { n , m } {\displaystyle k=\max\{n,m\}} . Each F I N k {\displaystyle \mathrm {FIN} _{k}} is a partial semigroup under + {\displaystyle +} . The tetris operation T : F I N k + 1 → F I N k {\displaystyle T\colon \mathrm {FIN} _{k+1}\to \mathrm {FIN} _{k}} is defined T ( f ) ( n ) = max { 0 , f ( n ) − 1 } {\displaystyle T(f)(n)=\max\{0,f(n)-1\}} . Intuitively, if f {\displaystyle f} is represented as a pile of square blocks, where the n {\displaystyle n} th column has height f ( n ) {\displaystyle f(n)} , then T ( f ) {\displaystyle T(f)} is the result of removing the bottom row. The name is in analogy with the video game. T ( k ) {\displaystyle T^{(k)}} denotes the k {\displaystyle k} th iterate of T {\displaystyle T} . A block sequence ( f n ) {\displaystyle (f_{n})} in F I N k {\displaystyle \mathrm {FIN} _{k}} is one such that max ( supp ( f m ) ) < min ( supp ( f m + 1 ) ) {\displaystyle \max(\operatorname {supp} (f_{m}))<\min(\operatorname {supp} (f_{m+1}))} for every m {\displaystyle m} .
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