Can't find such a tick-box anywhere (which is a fairly good clue that I've never meddled with it - if indeed there is one).
The default behaviour (at least in my experience) is you stay logged in unless you ask to log out - there is a button for the latter.
I found one of their own help pages which said a common reason for this problem is incorrect date and time settings on your computer. But they're not incorrect! It also said to clear cookies (tried that), and finally gave instructions to set LinkedIn as a trusted site in Chrome - which I did as well (even though I've never needed to in the past).
It still logs me out immediately on leaving the site.
I have to admit, I find the site pretty tiresome anyway - I don't use it to "sell myself", and I'm not terribly keen on people who do - there's only so much BS I can be bothered to read.
But it's gone from "mostly useless" to "almost completely useless", because it's not worth the nuisance of repeatedly signing in on the off-chance of some small thing that's not quite as dull, pretentious or irritating as the rest.
"John has spent seven years in the same job. Say congratulations to John!" Nooo, that's NOT an event.
"Sue has updated her skills: Microsoft Word." SHUT UP! We'll have people bragging they can read, next.