If we could see all the intersections, down to the point of very small groups (5-10 people), what would it look like?
If each category had two values uniformly distributed, then 29 categories would be more than enough to categorize each American individually.
How many categories non-uniformly distributed will it take?
A visualization for a few categories: https://cran.r-project.org/web/packages/nVennR/vignettes/nVennR.html
We would need something different for higher dimensions!
The data to answer this question might not exist.
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