A dichotomy on Schreier sets

A Dichotomy on Schreier Sets

We show that the Schreier sets Sα (α < ω1) satisfy the following dichotomy property. For every hereditary collection F of finite subsets of N, either there exists infinite M = (mi)1 ⊆ N such that Sα(M) = {{mi : i ∈ E} : E ∈ Sα} ⊆ F , or there exist infinite M = (mi)1 , N ⊆ N such that F [N ](M) = {{mi : i ∈ F} : F ∈ F and F ⊂ N} ⊆ Sα.

Polynomials on Schreier ’ s space

We introduce a weakened version of the Dunford-Pettis property, and give examples of Banach spaces with this property. In particular, we show that every closed subspace of Schreier’s space S enjoys it. As an application, we characterize the weak polynomial convergence of sequences, show that every closed subspace of S has the polynomial Dunford-Pettis property of Biström et al. and give other p...

On Schreier Varieties of Linear Algebras

Sharpness of KKL on Schreier graphs

Recently, the Kahn–Kalai–Linial (KKL) Theorem on influences of functions on {0, 1} was extended to the setting of functions on Schreier graphs. Specifically, it was shown that for an undirected Schreier graph Sch(G,X,U) with log-Sobolev constant ρ and generating set U closed under conjugation, if f : X → {0, 1} then E [f ] & log(1/M[f ]) · ρ ·Var[f ]. Here E [f ] denotes the average of f ’s inf...

Some remarks on James–Schreier spaces

The James–Schreier spaces Vp, where 1 6 p < ∞, were recently introduced by Bird and Laustsen [5] as an amalgamation of James’ quasi-reflexive Banach space on the one hand and Schreier’s Banach space giving a counterexample to the Banach–Saks property on the other. The purpose of this note is to answer some questions left open in [5]. Specifically, we prove that (i) the standard Schauder basis f...