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It is fundamental in number theory to calculate lower bounds for heightfunctions. Grizzard studied lower bounds for the Weil height in a relativesetting. Vidaux and Videla introduced the Northcott number for a set$A\subset\bar{\mathbb{Q}}$. It bounds the Weil height on $A$ from below,outside the zero-height points and the finitely many small-height points.Pazuki, Technau, and Widmer introduced the weighted Weil heights. These heightsgeneralize both the absolute and relative Weil heights. In this paper, weintroduce a relative version of the Northcott number related to the weightedWeil height. We also give a field extension whose Northcott number equals agiven positive number. The work is a relative version of the previous work ofthe author and Sano on the Northcott numbers for the weighted Weil heights.

Relative Northcott numbers for the weighted Weil heights