My interviewer mailed me this:
When can you come in for a technical interview?
Is it correct to say "15th June 2013 or any other Saturday after that" or is there any other correct phrase to say that?
You could say "the 15th of June 2013 or any other saturday after that" if you mean that you could attend on the 15th or, any saturday that is after that.
If you write the date the English and not American way, you would need the words the and of in it. Including the year is not necessary if you are talking about a date in the same year. If the 15th of June will be later in the same year, it would be sufficient to say "the fifteenth of June".
I don't see anything wrong with the phrase you are using. As alternative, you could say "any saturday, starting from June 15." (I used the American date format.)
The year is normally not necessary, if you are referring to the current year. In the case being explicit is necessary, you can say "any saturday, starting from June 15, 2013."