Is there a free version of Red-Hat Linux for home desktops?

Is there a free edition of Red-Hat for home desktops?

Yes, it's called Fedora.

There is also CentOS if you are still looking for a server environment. CentOS is compiled from Red Hat's source, and is essentially the same, just without any Red Hat branding. There are some minor differences, but the distro attempts to maintain 100% binary compatibility with Red Hat. You can download the latest release online for free.

Scientific Linux is another RHEL clone similar to CentOS, maintained by Fermilab and CERN for the High-Energy Physics community but usable by anyone.

Fedora is free Linux distribution sponsored by Red Hat. I think it's the closest you'll find.