I have some tags which are 2 or 3 characters but the search does not find them at all. And also 2 or 3 character strings don't match either even if they are full words in the title of a question.
Is there a way to make these found in the search?
Yes, in the "live" settings (Data entry section) adjust "Minimum length of search term for Ajax search". The default value is 4 characters.
It also will be necessary to configure your database accordingly. Btw, Postgresql has much better support for full text search - strongly recommend to switch your site from MySQL to Postgres, there are other very compelling reasons to switch as well - the sooner the better because migrating a small data set is much easier.
When you have time, please take a look at http://www.postgresql.org/docs/8.3/static/textsearch.html - postgres supports synonyms, thesaurus dictionaries, etc. things that allow fine tuning the search to suite your needs.
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