Speaking in relation to the UK, a litigant would only be considered to be vexatious where a claim was considered to be without merit and was only being brought for the purpose of causing annoyance, which is a fairly high threshold to prove.

With IV, the issue is that they at least have an arguable case that patents they hold have been infringed. I would therefore think that they would not approach the threshold of being considered a vexatious litigant.

The problem is that the patents have been granted in the first place, and once granted, companies are free to use them as they please or at least argue their case. Of course, the same patents could be revoked in a court action but I acknowledge that the principal aim is to extract royalties from companies before the matter ever comes to court.

On the California provision you mention, this contains certain caveats which would mean the person would be free to file as many suits as they wanted where the claim was of a sufficiently low value. Presumably the provision is designed to prevent the individual representing themself from continuously filing frivolous suits against companies claiming vast damages, arguably due to the chilling effect this might have on companies doing business.

The provision you're referring to only applies to pro se litigants (people who represent themselves in court).

A litigant though is a 'legal person'. if you have 2000 legal entities (ie, legally== persons), it seems to help you on the math. But your question still stands and is a good one. Its not clear what role a common "controling entity" would have, for example, on the ability of affiliates, spv's and relatives/children to file or be found 'vexatious' etc.[1]

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[1] INAL, but I think this question should be answerable.

While I'm generally against RICO as law, this seems like one case where it would be reasonable to apply.