%0 Journal Article
%T Positional graphs and conditional structure of weakly null sequences
%A J. Lopez-Abad
%A S. Todorcevic
%J Mathematics
%D 2011
%I arXiv
%X 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 $p\ge 1$.
%U http://arxiv.org/abs/1111.5150v1