Minimizing Indexed Reads with Complex Criteria

Since you are selecting fields from Table1 only, it is pointless to get the data from a join applied on table 1 and table 2. After testing both queries on my server, i have noticed that the processing time has reduced using the first query (nested selects). hope this answers your question.

This can be improved in a thousand and one ways, then it should be a matter of milliseconds.

Better Queries

This is just your query reformatted with aliases and some noise removed to clear the fog:

SELECT count(DISTINCT t.id)
FROM   tickets      t
JOIN   transactions tr ON tr.objectid = t.id
JOIN   attachments  a  ON a.transactionid = tr.id
WHERE  t.status <> 'deleted'
AND    t.type = 'ticket'
AND    t.effectiveid = t.id
AND    tr.objecttype = 'RT::Ticket'
AND    a.contentindex @@ plainto_tsquery('frobnicate');

Most of the problem with your query lies in the first two tables tickets and transactions, which are missing from the question. I'm filling in with educated guesses.

• t.status, t.objecttype and tr.objecttype should probably not be text, but enum or possibly some very small value referencing a look-up table.

EXISTS semi-join

Assuming tickets.id is the primary key, this rewritten form should be much cheaper:

SELECT count(*)
FROM   tickets t
WHERE  status <> 'deleted'
AND    type = 'ticket'
AND    effectiveid = id
AND    EXISTS (
   SELECT 1
   FROM   transactions tr
   JOIN   attachments  a ON a.transactionid = tr.id
   WHERE  tr.objectid = t.id
   AND    tr.objecttype = 'RT::Ticket'
   AND    a.contentindex @@ plainto_tsquery('frobnicate')
   );

Instead of multiplying rows with two 1:n joins, only to collapse multiple matches in the end with count(DISTINCT id), use an EXISTS semi-join, which can stop looking further as soon as the first match is found and at the same time obsoletes the final DISTINCT step. Per documentation:

The subquery will generally only be executed long enough to determine whether at least one row is returned, not all the way to completion.

Effectiveness depends on how many transactions per ticket and attachments per transaction there are.

Determine order of joins with join_collapse_limit

If you know that your search term for attachments.contentindex is very selective - more selective than other conditions in the query (which is probably the case for 'frobnicate', but not for 'problem'), you can force the sequence of joins. The query planner can hardly judge selectiveness of particular words, except for the most common ones. Per documentation:

join_collapse_limit (integer)

[...]
Because the query planner does not always choose the optimal join order, advanced users can elect to temporarily set this variable to 1, and then specify the join order they desire explicitly.

Use SET LOCAL for the purpose to only set it for the current transaction.

BEGIN;
SET LOCAL join_collapse_limit = 1;

SELECT count(DISTINCT t.id)
FROM   attachments  a                              -- 1st
JOIN   transactions tr ON tr.id = a.transactionid  -- 2nd
JOIN   tickets      t  ON t.id = tr.objectid       -- 3rd
WHERE  t.status <> 'deleted'
AND    t.type = 'ticket'
AND    t.effectiveid = t.id
AND    tr.objecttype = 'RT::Ticket'
AND    a.contentindex @@ plainto_tsquery('frobnicate');

ROLLBACK; -- or COMMIT;

The order of WHERE conditions is always irrelevant. Only the order of joins is relevant here.

Or use a CTE like @jjanes explains in "Option 2". for a similar effect.

Indexes

B-tree indexes

Take all conditions on tickets that are used identically with most queries and create a partial index on tickets:

CREATE INDEX tickets_partial_idx
ON tickets(id)
WHERE  status <> 'deleted'
AND    type = 'ticket'
AND    effectiveid = id;

If one of the conditions is variable, drop it from the WHERE condition and prepend the column as index column instead.

Another one on transactions:

CREATE INDEX transactions_partial_idx
ON transactions(objecttype, objectid, id)

The third column is just to enable index-only scans.

Also, since you have this composite index with two integer columns on attachments:

"attachments3" btree (parent, transactionid)

This additional index is a complete waste, delete it:

"attachments1" btree (parent)

Details:

• Is a composite index also good for queries on the first field?

GIN index

Add transactionid to your GIN index to make it a lot more effective. This may be another silver bullet, because it potentially allows index-only scans, eliminating visits to the big table completely.
You need additional operator classes provided by the additional module btree_gin. Detailed instructions:

• Inner join using an array column

"contentindex_idx" gin (transactionid, contentindex)

4 bytes from an integer column don't make the index much bigger. Also, fortunately for you, GIN indexes are different from B-tree indexes in a crucial aspect. Per documentation:

A multicolumn GIN index can be used with query conditions that involve any subset of the index's columns. Unlike B-tree or GiST, index search effectiveness is the same regardless of which index column(s) the query conditions use.

Bold emphasis mine. So you just need the one (big and somewhat costly) GIN index.

Table definition

Move the integer not null columns to the front. This has a couple of minor positive effects on storage and performance. Saves 4 - 8 bytes per row in this case.

                      Table "public.attachments"
         Column      |            Type             |         Modifiers
    -----------------+-----------------------------+------------------------------
     id              | integer                     | not null default nextval('...
     transactionid   | integer                     | not null
     parent          | integer                     | not null default 0
     creator         | integer                     | not null default 0  -- !
     created         | timestamp                   |                     -- !
     messageid       | character varying(160)      |
     subject         | character varying(255)      |
     filename        | character varying(255)      |
     contenttype     | character varying(80)       |
     contentencoding | character varying(80)       |
     content         | text                        |
     headers         | text                        |
     contentindex    | tsvector                    |