A characterization of third Engel groups (Q1893233)

The author gives a characterization of \(3\)-Engel groups by means of a combinatorial property. She defines the class \({{\mathcal E}_ 3}^*\) of those groups \(G\) satisfying the following condition: For every pair of infinite subsets \(X\), \(Y\) of \(G\), there exist \(x\) in \(X\) and \(y\) in \(Y\) such that \([x, y, y, y] = 1\) and proves that any infinite group in the class \({{\mathcal E}_ 3}^*\) is a third Engel group.