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
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.