Erdős Problem #1219

Let \((n_k)\) be an increasing sequence of integers such that \(2^{\aleph_{n_k}}\) is strictly increasing, and \(2^{\aleph_{n_0}} > \aleph_\omega\). Is it true that \[\sum_k 2^{\aleph_{n_k}} \to (\aleph_\omega)^2?\] Here \(\sum_k 2^{\aleph_{n_k}}\) is the cardinal sum, and the partition relation is the classical one with two colours: every \(2\)-colouring of the pairs of a set of that cardinality has a monochromatic set of cardinality \(\aleph_\omega\). A problem of Erdős, Hajnal, and Rado [ErHa71], proved by Shelah [Sh75].