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Recently, Schlicht and Stephan lifted the notion of automatic-structures tothe notion of (finite-word) ordinal-automatic structures. These are structureswhose domain and relations can be represented by automata reading finite wordswhose shape is some fixed ordinal $\alpha$. We lift Delhomm\'e'srelative-growth-technique from the automatic and tree-automatic setting to theordinal-automatic setting. This result implies that the random graph is notordinal-automatic and infinite integral domains are not ordinal-automatic withrespect to ordinals below $\omega_1+\omega^\omega$ where $\omega_1$ is thefirst uncountable ordinal.

The field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic