The thing is, there's no single day when the whole country celebrates, because there are many patron saints. Each family has their own: men inherit theirs, and women usually celebrate their father's saint first, then their husband's (though sometimes after marriage they celebrate both twice a year).
This tradition isn't unique to Serbia—you'll find it in neighboring Balkan countries too. Though Serbians often see Slava as a kind of marker of their national identity. But honestly, it's more complicated than that.
PS. For 2023, I counted about 10 public holidays total, plus you get 20 vacation days (though I know people whose employers give them 25). Guess I should read through the Serbian labor code again—maybe there are some other perks hiding in there :()
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