Down-up matchings in infinite graphs

A down-up matching in an ordinal graph is a matching of a set \( A\) with a set \( B\) where every element of \( A\) is less than all elements of \( B\). Suppose for every graph on an ordinal \( \alpha\), there is either an independent set of order type \( \beta\) or a down-up matching from a set \( A\) to a set \( B\) and \( A\) has order type \( \gamma\). Then we write \( \alpha\rightarrow(\beta,\gamma-\)matching\( )^2\).

A problem on ordinal graphs and down-up matchings (proposed by Erdös and Larson [1])
Suppose that \( j\) and \( k\) are positive integers with \( k\ge 2\) and \( \eta\) is a limit ordinal. Is it true that \( \omega^{\eta+jk}\rightarrow(\omega^{\eta+j},\omega^k-\)matching\( )^2\)?

If \( j\) and \( k\geq 2\) are positive integers and \( \eta\) is a countable limit ordinal, then Erdös and Larson [1] \( \omega^{\eta+jk+1}\rightarrow (\omega^{\eta+j},\gamma-\)matching\( )^2\) but \( \omega^{\eta+jk-1}\not\rightarrow (\omega^{\eta+j},\gamma-\)matching\( )^2\).



Bibliography
1 P. Erdös and J. A. Larson, Matchings from a set below to a set above, Directions in infinite graph theory and combinatorics (Cambridge, 1989), Discrete Math. 95 (1991) no. 1-3, 169-182.