Make host and link template to that host.
On the template create 1 item, one trigger and force the trigger to go into the problem state.

Verify on Monitoring -> problems with the appropriate filters.

Now, go back to configuration -> hosts and unlink and clear the template.

Verify on Monitoring -> problems with the appropriate filters and realise that all problems that happened on the host are gone as well, by just deleting the template(trigger).

The moment the trigger is removed, the problem is not presented anymore., while the event remains in the DB.

I think this is unwanted behaviour, the problem was considered a problem at the time, but it does not mean that if the trigger is removed, it was never a problem. I think the problems should be stored and presented as per configuration of the housekeeper process.