Calendar issue on OS X Mavericks
It would appear that the upgrade has failed to update your Calendar app. OS X 10.9 has Calendar v7.0 and if you do have 6.0 then something went wrong.

I would suspect that there is some serious problem with your existing Calendar app such as permissions.

Delete your Calendar app and either grab a copy from another Mavericks Mac or re-install 10.9.

If you go the re-install route I would be tempted to do a reformat and clean install myself, just to make sure there are no hidden gotchas since you already have one. At the least I would boot into recovery mode and run "Fix Permissions" before the re-install.