Can PostgreSQL index array columns?

Yes you can index an array, but you have to use the array operators and the GIN-index type.

Example:

    CREATE TABLE "Test"("Column1" int[]);
    INSERT INTO "Test" VALUES ('{10, 15, 20}');
    INSERT INTO "Test" VALUES ('{10, 20, 30}');

    CREATE INDEX idx_test on "Test" USING GIN ("Column1");

    -- To enforce index usage because we have only 2 records for this test... 
    SET enable_seqscan TO off;

    EXPLAIN ANALYZE
    SELECT * FROM "Test" WHERE "Column1" @> ARRAY[20];

Result:

Bitmap Heap Scan on "Test"  (cost=4.26..8.27 rows=1 width=32) (actual time=0.014..0.015 rows=2 loops=1)
  Recheck Cond: ("Column1" @> '{20}'::integer[])
  ->  Bitmap Index Scan on idx_test  (cost=0.00..4.26 rows=1 width=0) (actual time=0.009..0.009 rows=2 loops=1)
        Index Cond: ("Column1" @> '{20}'::integer[])
Total runtime: 0.062 ms

Note

it appears that in many cases the gin__int_ops option is required

create index <index_name> on <table_name> using GIN (<column> gin__int_ops)

I have not yet seen a case where it would work with the && and @> operator without the gin__int_ops options

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