(Mishnah): Some lost articles may be kept and some must be announced...

He may keep the following items (since the owner would have given up hope of retrieving them) - Scattered produce, scattered money, bundles in the public domain, fig circles, bakery-made loaves, strings of fish, pieces of meat, shearings of unprocessed wool, bundles of linen thread and tongues of scarlet wool.

(Gemara - question): What is the Torah source for (allowing a person to acquire when) the owner gave up hope?

R. Yochanan citing R. Shimon ben Yehutzedek: The pasuk states (Devarim 20, 2), "so you shall do for his donkey...(that shall be lost from him)'' - you must announce any item that is lost from him and that you found, but if it is not lost from him, you do not need to announce it. This excludes an item for which the owner gave up hope, that it is lost from him and from everyone else too (such as if it was washed away by the river, as certainly the owner gave up hope of retrieving it).