Positional graphs and conditional structure of weakly null sequences

We prove that, unless assuming additional set theoretical axioms, there are no reflexive spaces without unconditional sequences of the density continuum. We show that for every integer nn there are normalized weakly-null sequences of length ωnωn without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that ωωωω is the minimal cardinal κκ that could possibly have the property that every weakly null κκ-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κκ which is smaller than the first ωω-Erdős 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 c0c0 or ℓpℓp, with p≥1p≥1.