I'm trying to make a directory where all users can write, but no user can delete another users file in that directory.


You want a directory where the t-bit is set, such as /tmp/, /var/tmp.

In the chmod manual page, this is referred to as the "sticky bit":

The restricted deletion flag or sticky bit is a single bit, whose interpretation depends on the file type. For directories, it prevents unprivileged users from removing or renaming a file in the directory unless they own the file or the directory; this is called the restricted deletion flag for the directory, and is commonly found on world-writable directories like /tmp.

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