What is the minimum size of the partition for a hard disk?

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The internal hardware parts of a computer are often referred to as components, while external hardware devices are usually called peripherals. Together, they all fall under the category of computer hardware. Software, on the other hand, consists of the programs and applications that run on computers. Because software runs on computer hardware, software programs often have system requirements that list the minimum hardware required for the software to run.

The minimum size of the partition for a hard disk ................

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32k using FAT 12.

well, a hard disk manager program can create a partition that has near zero size -- but it would be useless [except to 'hide' security files]

in what file system? depends on what partition allocation size you set, and options vary based on file system.

It depends on the filesystem but generally, you can make it as small as you want or fill up the entire hard disk.

I usually make at least 3 partitions, one for the OS and programs (so they don't mess up with my data), one for files (for easier recovery and safer system upgrades), and one for virtual memory (since I can't afford that much RAM).

The minimum size for a partition is based on the purpose of that partition.  Many OEM Hard Drives have a "Recovery" partition set aside for the storage of System Recovery files, and in some cases I think it is as small as 1 GB (Gigabyte).  50 or 100 GBs might make a bit more sense from a practical standpoint, given the storage requirements of modern Operating Systems including those of Microsoft Corporation and the many, many dozens or hundreds of "flavors" of the Linux O.S.

A partition is simply a set aside space, so without more context, your question reads a bit like asking, 'What is the minimum size of a Spare bedroom on a house?'  Is it going to have only 1 spare bed, a Twin bed, or does it need a small desk, a dresser and "room" to turn around while doing the Hokey Pokey?

zero. you can make the hard drive one partition.

What file system ? It's about 10 MB for NTFS.