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 n$n$ there are normalized weakly-null sequences of length _ωn${\omega }_n$ without unconditional subsequences. This together with a result of Dodos et al. (2011) [7] shows that _ωω${\omega }_{\omega }$ is the minimal cardinal κ$\kappa$ that could possibly have the property that every weakly null κ$\kappa$-sequence has an infinite unconditional basic subsequence. We also prove that for every cardinal number κ$\kappa$ which is smaller than the first ω$\omega$-Erd?s 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 _c0$c_0$ or _?p${?}_p$, with p≥1$p\ge 1$.     

Consistency,optimality, and incompleteness

Assume that the problem _P0$P_0$ is not solvable in polynomial time. Let T   be a first-order theory containing a sufficiently rich part of true arithmetic. We characterize T∪{_ConT}$T\cup \{{\mathit{Con}}_T\}$ as the minimal extension of T   proving for some algorithm that it decides _P0$P_0$ as fast as any algorithm B$\mathbb{B}$ with the property that T   proves that B$\mathbb{B}$ decides _P0$P_0$. Here, _ConT${\mathit{Con}}_T$ claims the consistency of T. As a byproduct, we obtain a version of Gödel?s Second Incompleteness Theorem. Moreover, we characterize problems with an optimal algorithm in terms of arithmetical theories.