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> There would likely be a lot of unaffiliated candidates if the number in the House was drastically and suddenly increased.

No, there wouldn't. The electoral forces that drive duopoly operate the same almost independently of district size, changing the number of single-member House districts with first-past-the-post elections won't reduce duopoly.

Changing the electoral method might (and doesn't require changing the number of districts.)

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I don't agree with this premise. From what I learned about redistricting when I was a candidate for California's citizen redistricting board (which almost completely eliminated gerrymandering in CA because legislature no longer has authority to define districts), under-representation is a big issue. From my impression, Republican and Democratic party bases are much smaller than everyone thinks. Neither of them would be in control of a drastically increased body without a collation with many many other groups. Also, many laws are currently in place that prevent third parties from participating in the primary system on equal footing with Dems and Rep. If those limitations where removed, that by itself would see an erosion of Dem and Rep party affiliation. The hold that Dems and Reps have on US politics is much more tenuous than they'd like everyone to believe.

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