In mathematics, Gowers' theorem, also known as Gowers' Ramsey theorem and Gowers' FIN"k" 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 formula_1, the support of formula_2 is defined formula_3. Given formula_4, let formula_5 be the set
 formula_6
If formula_7, formula_8 have disjoint supports, we define formula_9 to be their pointwise sum, where formula_10. Each formula_5 is a partial semigroup under formula_12.
The "tetris operation" formula_13 is defined formula_14. Intuitively, if formula_2 is represented as a pile of square blocks, where the formula_16th column has height formula_17, then formula_18 is the result of removing the bottom row. The name is in analogy with the video game. formula_19 denotes the formula_20th iterate of formula_21.
A "block sequence" formula_22 in formula_5 is one such that formula_24 for every formula_25.
The theorem.
Note that, for a block sequence formula_22, numbers formula_27 and indices formula_28, the sum formula_29 is always defined. Gowers' original theorem states that, for any finite colouring of formula_5, there is a block sequence formula_22 such that all elements of the form formula_29 have the same colour.
The standard proof uses ultrafilters, or equivalently, nonstandard arithmetic.
Generalisation.
Intuitively, the tetris operation can be seen as removing the bottom row of a pile of boxes. It is natural to ask what would happen if we tried removing different rows. Bartošová and Kwiatkowska considered the wider class of "generalised tetris operations", where we can remove any chosen subset of the rows.
Formally, let formula_33 be a nondecreasing surjection. The induced tetris operation formula_34 is given by composition with formula_35, i.e. formula_36. The generalised tetris operations are the collection of formula_37 for all nondecreasing surjections formula_33. In this language, the original tetris operation is induced by the map formula_39.
Bartošová and Kwiatkowska showed that the finite version of Gowers' theorem holds for the collection of generalised tetris operations. Lupini later extended this result to the infinite case.