Normal sequences without strongly normal subsequences

A sequence $(y_n^{\ast })$ in the unit sphere of $E^{\ast }$ is strongly normal if the vectors $x\in E$ satisfying ${\sum }_n\mathrm{\mid }{y}_n^{\ast }(x)\mathrm{\mid }<\mathrm{\infty }$ form a dense subspace. For every set $\mathrm{\Gamma }$ of cardinality at least the continuum, we construct a normalized weakly null sequence in ${\mathrm{\ell }}_1(\mathrm{\Gamma }{)}^{\ast }$ with no strongly normal subsequence. This answers a question of Śliwa negatively in ZFC. Together with the classical selection argument below the bounding number $\mathfrak{b}$, the construction gives positive and negative bounds for this subsequence property on ${\mathrm{\ell }}_1(\kappa )$. Its validity on ${\mathrm{\ell }}_1({\omega }_1)$ is independent of ZFC. The intermediate range $\mathfrak{b}⩽\kappa <\mathfrak{c}$ remains undecided by these results. We also give a counterexample in a Banach sequence space that is not isomorphic to any ${\mathrm{\ell }}_1(\mathrm{\Gamma })$.