Postgresql – the optimal data type for an MD5 field

For Postgres 9.1 or later:

CREATE INDEX idx_time_limits_ts_inverse
ON time_limits (id_phi, start_date_time, end_date_time DESC);

In most cases the sort order of an index is hardly relevant. Postgres can scan backwards practically as fast. But for range queries on multiple columns it can make a huge difference. Closely related:

• PostgreSQL index not used for query on range

Consider your query:

SELECT *
FROM   time_limits
WHERE  id_phi = 0
AND    start_date_time <= '2010-08-08 00:00'
AND    end_date_time   >= '2010-08-08 00:05';

Sort order of the first column id_phi in the index is irrelevant. Since it's checked for equality (=), it should come first. You got that right. More in this related answer:

• Multicolumn index and performance

Postgres can jump to id_phi = 0 in next to no time and consider the following two columns of the matching index. These are queried with range conditions of inverted sort order (<=, >=). In my index, qualifying rows come first. Should be the fastest possible way with a B-Tree index¹:

• You want start_date_time <= something: index has the earliest timestamp first.
• If it qualifies, also check column 3.
  Recurse until the first row fails to qualify (super fast).
• You want end_date_time >= something: index has the latest timestamp first.
• If it qualifies, keep fetching rows until the first one doesn't (super fast).
  Continue with next value for column 2 ..

Postgres can either scan forward or backward. The way you had the index, it has to read all rows matching on the first two columns and then filter on the third. Be sure to read the chapter Indexes and ORDER BY in the manual. It fits your question pretty well.

How many rows match on the first two columns?
Only few with a start_date_time close to the start of the time range of the table. But almost all rows with id_phi = 0 at the chronological end of the table! So performance deteriorates with later start times.

Planner estimates

The planner estimates rows=62682 for your example query. Of those, none qualify (rows=0). You might get better estimates if you increase the statistics target for the table. For 2.000.000 rows ...

ALTER TABLE time_limits ALTER start_date_time SET STATISTICS 1000;
ALTER TABLE time_limits ALTER end_date_time   SET STATISTICS 1000;

... might pay. Or even higher. More in this related answer:

• Check statistics targets in PostgreSQL

I guess you don't need that for id_phi (only few distinct values, evenly distributed), but for the timestamps (lots of distinct values, unevenly distributed).
I also don't think it matters much with the improved index.

CLUSTER / pg_repack / pg_squeeze

If you want it faster, yet, you could streamline the physical order of rows in your table. If you can afford to lock your table exclusively (at off hours for instance), rewrite your table and order rows according to the index with CLUSTER:

CLUSTER time_limits USING idx_time_limits_inversed;

Or consider pg_repack or the later pg_squeeze, which can do the same without exclusive lock on the table.

Either way, the effect is that fewer blocks need to be read from the table and everything is pre-sorted. It's a one-time effect deteriorating over time with writes on the table fragmenting the physical sort order.

GiST index in Postgres 9.2+

¹ With pg 9.2+ there is another, possibly faster option: a GiST index for a range column.

• There are built-in range types for timestamp and timestamp with time zone: tsrange, tstzrange. A btree index is typically faster for an additional integer column like id_phi. Smaller and cheaper to maintain, too. But the query will probably still be faster overall with the combined index.

• Change your table definition or use an expression index.

• For the multicolumn GiST index at hand you also need the additional module btree_gist installed (once per database) which provides the operator classes to include an integer.

The trifecta! A multicolumn functional GiST index:

CREATE EXTENSION IF NOT EXISTS btree_gist;  -- if not installed, yet

CREATE INDEX idx_time_limits_funky ON time_limits USING gist
(id_phi, tsrange(start_date_time, end_date_time, '[]'));

Use the "contains range" operator @> in your query now:

SELECT *
FROM   time_limits
WHERE  id_phi = 0
AND    tsrange(start_date_time, end_date_time, '[]')
    @> tsrange('2010-08-08 00:00', '2010-08-08 00:05', '[]')

SP-GiST index in Postgres 9.3+

An SP-GiST index might be even faster for this kind of query - except that, quoting the manual:

Currently, only the B-tree, GiST, GIN, and BRIN index types support multicolumn indexes.

Still true in Postgres 12.
You would have to combine an spgist index on just (tsrange(...)) with a second btree index on (id_phi). With the added overhead, I'm not sure this can compete.
Related answer with a benchmark for just a tsrange column:

• Perform this hours of operation query in PostgreSQL

get the most frequently appearing user_code where the month is specified

Since user_code is the primary key, that question would be nonsense. There can never be more than one. I assume you meant invite_code?

Just add a WHERE clause. And since the column can be NULL, also consider excluding NULL values:

SELECT invite_code, COUNT(*) AS counted
FROM   invite_table
WHERE  month = 'May'  -- or whatever is stored in your varchar(3) column
AND    invite_code IS NOT NULL -- exclude NULL
GROUP  BY invite_code
ORDER  BY counted DESC, invite_code  -- to break ties in deterministic fashion
LIMIT  10;

Month, date, timestamp?

A month column as varchar(3) doesn't seem very useful if there can be data for more than a single year. I would use data type date for it. You can format that with to_char() any way you like for presentation. Like:

SELECT to_char(date '2017-12-01', 'Mon');  -- 'Dec'

The column could look like this (also addressing your comment):

...
, inserted_at date DEFAULT CURRENT_DATE
...

The default value is entered when the column is omitted in an INSERT statement.

Or, if really only the month is relevant:

... DEFAULT date_trunc('month', now())::date

Or store the complete timestamptz (8 bytes, that's what I would probably do):

...
, inserted_at timestamptz DEFAULT now()
...

Read the manual here and here.

And be aware that date and timestamp depend on your current time zone setting. Details:

• Ignoring timezones altogether in Rails and PostgreSQL