This is an announcement for the paper "Positional graphs and conditional structure of weakly null sequences" by J. Lopez-Abad and S. Todorcevic.

Abstract: We prove that, unless assuming additional set theoretical axioms, there are no reflexive space without unconditional sequences of density the continuum. We give for every integer $n$ there are normalized weakly-null sequences of length $\om_n$ without unconditional subsequences. This together with a result of \cite{Do-Lo-To} shows that $\om_\omega$ is the minimal cardinal $\kappa$ that could possibly have the property that every weakly null $\kappa$-sequence has an infinite unconditional basic subsequence . We also prove that for every cardinal number $\ka$ which is smaller than the first $\om$-Erd"os cardinal there is a normalized weakly-null sequence without subsymmetric subsequences. Finally, we prove that mixed Tsirelson spaces of uncountable densities must always contain isomorphic copies of either $c_0$ or $\ell_p$, with $p\ge 1$.

Archive classification: math.FA math.LO

Mathematics Subject Classification: Primary 46B03, 03E35, Secondary 03E02, 03E55, 46B26, 46A35

Submitted from: abad@icmat.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1111.5150

or

http://arXiv.org/abs/1111.5150