What does the "503 day" refer to?
503 day is supposed to be the 3rd of May (5/03), which is the Edward x Winry day, or Edwin day in short.

There are two reasons for this coming into existence.

First, there's a jeans brand called Edwin and one of their logos contains the number 503.



Second, when Winry goes to Central City she's in room 503 of the hotel.