A bit of background: June 3rd was the original date of Lorelai and Luke's wedding, but then they called it off. At another character's wedding, Lorelai gives a drunken toast and keeps repeating the date. I searched for this quote online but never found it, so here it is from Season 6, Episode 19 "I Get a Sidekick Out of You," which originally aired on April 18, 2006:
"Hey do you know what date I'm not getting married? JUNE THIRD. Do not save the date. Do you hear me? Do whatever you want on June third, because there's nothing at all happening on that day. If there's anything you need to book or anything, it's totally safe to book it on JUNE THIRD. So, congratulations Lane and Zach. Who else here had 8 shots of tequila? Anybody? Hands? No."