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 $om$-Erdos 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 $pge 1$.
