In case you can't read the photo, it says that you can 'help someone do something' or you can 'help someone to do something'. So far, so familiar to me.
But then it goes on to say that know has the same pattern with
(1) Have you ever known them come on time?and
(2) Have you ever known them to come on time?
Now, if I have ever seen sentences of type (1) in the wild, I must have assumed them to have typos, because if I want to know someone/thing + verb, I must have the the to-infinitive form of the verb. Yes to (2), no to (1). Absolutely, no question.
So, I turned to the (English) Syntactician, who said that yes, (1) is good in her BrE, "but old-fashioned". I then went onto Twitter to proclaim my ignorance/learning/disbelief, and many English people (many of whom are probably not terribly old-fashioned) replied to say "Yes, that's fine. I can say that." No US people replied to say they could say it, and now that I look in Algeo's British or American English?, I see that he records this as a British form.
Palmer hasn't mentioned the big restriction on this, however. Algeo does, but I learned the restriction the hard way: by tweeting "Can you really know someone do something?" The answer there is 'no'--British English speakers can only use the to-less version in the perfect aspect (the 'have/had verbed' forms). So:
- General (BrE or AmE) perfect: I have known them to frequent dark alleys.
- BrE-only perfect: I have known them frequent dark alleys.
- General English present: I know them to frequent dark alleys.
- Nobody's English present: *I know them frequent dark alleys.
(Overly academic side point. Skip this unless can name at least two theories of grammar!
I'm wondering how you get a [say, Chomskyan] theory of grammar to account for a complementation structure that is particular to a certain aspect of a certain verb. Maybe all theories are now so lexical that it's possible--though you'd have to treat known and know as different lexical items, I guess. Would be easier to account for in a Construction Grammar, but still seems like a very heavy--or at least fiddly--cognitive load for a language to bear. If you know about such things, let me know in the comments, please!)
I should also say a bit about that help (to). As I said above, both of these are fine in AmE and BrE:
...but what's interesting for us is that AmE prefers (3) [in 75% of the cases in the Brown corpus] and BrE prefers (4) [73% of the cases in the LOB corpus] (both figures from Algeo, p. 228).
(3) I helped them escape.
(4) I helped them to escape.
And that, my friends, is how you write a blog post of less than 1000 words. When was the last time you had known me do that? :)