How to get maximum number of concurrent events in postgresql?

I have a table named events like this:

id: int
source_id: int
start_datetime: timestamp
end_datetime: timestamp  

These events could have overlaps, and I want to know maximum number of overlapping events that have occurred over a time span. For example, in a situation like this:

id | source_id | start_datetime     | end_datetime
----------------------------------------------------------
1  | 23        | 2017-1-1T10:20:00  | 2017-1-1T10:40:00
1  | 42        | 2017-1-1T10:30:00  | 2017-1-1T10:35:00
1  | 11        | 2017-1-1T10:37:00  | 2017-1-1T10:50:00  

The answer is 2, because at most 2 events overlap at 10:30 until 10:35.
I’m using Postgres 9.6

Answer

Here is the idea: count the number of starts and subtract the number of stops. That gives the net amount at each time. The rest is just aggregation:

with e as (
      select start_datetime as dte, 1 as inc
      from events
      union all
      select end_datetime as dte, -1 as inc
      from events
     )
select max(concurrent)
from (select dte, sum(sum(inc)) over (order by dte) as concurrent
      from e
      group by dte
     ) e;

The subquery shows the number of overlapping events at each time.

You can get the time frame as:

select dte, next_dte, concurrent
from (select dte, sum(sum(inc)) over (order by dte) as concurrent,
             lead(dte) over (partition by dte) as next_dte
      from e
      group by dte
     ) e
order by concurrent desc
fetch first 1 row only;