Does Erase Disk delete the Recovery Partition? I was trying to reformat my Mac Mini Mid-2011, so I went to Disk Utility. I saw that there was a Recovery Partition. Then I did an Erase Disk, which I thought was only erased the main partition and would leave the Recovery Partition.

Then I rebooted and held down cmd+R. The screen showed "Starting Internet Recovery", which means that my local Recovery Partition is gone, right? :(

I was hoping to save time by using the local Recovery Partition instead of downloading from internet.
When you want to reformat there are two things you can do, reformat the partition or reformat the drive. Or more properly "Re-partitioning."

It is important when you don't want to do what you did (repartition the whole drive -or perhaps reformat both, not sure-) to select the "Macintosh HD" entry (or whatever your HD is called) on the left side of Disk Utility. 

Note that each drive has (at least) two entries. The physical drive and the partition (or partitions). Apple usually does not show the recovery or boot partition (grrr....). And if the discovery triangle is pointing right instead of down then you just see the drive and not the enclosed partition.

You can open disk utility safely and just look to see which is which and safely select either entry to see what options are available for each. The differences are subtle.