Summary: The following graph-drawing problems are known to be complete for the existential theory of the reals (\(\exists \mathbb{R}\))-complete) as long as the parameter \(k\) is unbounded. Do they remain \(\exists \mathbb{R}\)-complete for a fixed value \(k\)?

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Do \(k\) graphs on a shared vertex set have a simultaneous geometric embedding?

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Is \(G\) a segment intersection graph, where \(G\) has maximum degree at most \(k\)?

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Given a graph \(G\) with a rotation system and maximum degree at most \(k\), does \(G\) have a straight-line drawing which realizes the rotation system?

We show that these, and some related, problems remain \(\exists \mathbb{R}\)-complete for constant \(k\), where \(k\) is in the double or triple digits. To obtain these results we establish a new variant of Mnëv’s universality theorem, in which the gadgets are placed so as to interact minimally; this variant leads to fixed values for \(k\), where the traditional variants of the universality theorem require unbounded values of \(k\).