Finite forms of Gowers' Theorem on the oscillation stability of c₀

2013/12/17 by Diana Ojeda-Aristizabal, Ojeda-Aristizabal, Diana
Mathematics · #05D10 #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #math.CO #math.FA #msc:05D10

paper · pdf · doi:10.48550/arxiv.1312.4639

18 pages

arxiv created 2013/12/17 · arxiv updated 2013/12/18

Abstract

We give a constructive proof of the finite version of Gowers' FIN_k Theorem and analyse the corresponding upper bounds. The FIN_k Theorem is closely related to the oscillation stability of c₀. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was studied well before by V. Milman. We compare the finite FIN_k Theorem with the finite stabilization principle in the case of spaces of the form ℓ_∞ⁿ, n∈ℕ and establish a much slower growing upper bound for the finite stabilization principle in this particular case.

Related

• Another proof of a Gowers theorem
• An infinite-dimensional version of Gowers' \FIN_{\± k}\n theorem
• A lower bound on Gowers' FIN_k theorem
• Combinatorial proofs of Gowers' FIN_k and FIN_k^± theorem
• Oscillation stability by the Carlson-Simpson theorem