Which is a better database to use for Askbot, PostgreSQL or MySQL, and why?

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PostgresQL is better because it supports transactions and full text search at the same time.

MySQL does not support both at once. MySQL has a few choices of data storage backends - and none of the backends supports those two things at simultaneously.

Postgres has very deep support of full text searching - including word stemming, thesaurus matching, ranking, etc.

Transactions help to keep data internally consistent and avoid race conditions - when two or more users read and write data simultaneously.

P.S. This site still runs or MySQL, my bad. If you use Postgres, the only visible change will be "sort by relevance" option in the question list, that will appear when some search term is typed. MySQL does not have ranking, so no such option.