I've seen Niels recommending Slack 8.1, or at least Slack 7.1 for those ancient computers. But for 386sx with 6mb of RAM you migh try one of old (ancient) versions.
Slack 4.0 is pretty much obsolete: it uses libc5 and Linux 2.0.x. But it's more lightweight. The biggest problem with it is that you won't be able to compile any fairly more complex modern program. Basic Linux 3.5 is based on 4.0 and has some more modern software available, although I'm not sure if much of those will work with only 6 mb of ram. They also use Tiny X instead of XFree86 (3.x is default in Slack 4). Don't know what are minimal requirements for that. I would think 8 mb of ram.
Slackware 2.3 used Linux 1.3 and libc4. It's pretty much archeological stuff, but will probably run pretty fast on those oldies.
The software, on the other side might be bigger problem than with 4.0. It also comes with an ancient version of gcc (2.6). My testing in qemu showed that it will work with 2-3 mb of RAM (without X). Install requires 4 mb of RAM or a swap partiton. You would need at least 8 mb for X (XFree86 2.x - X11R5), but it might work with only 6 mb. I guess you would need swap for that. But it will be slow as heck. There are still software that might run fine (mostly xaw stuff).
You might take a look at retro page at Niels Horn's site:
http://www.nielshorn.net/slackware/slack_old.php. Those are run under QEmu and vmWare, but might contain some tips.
In any case I would try Slack 7.1, or try slimming 8.1 (turning off unnecessary stuff in rc scripts and go through the stuff in rc.modules to turn off unneeded things) before that.
Btw, Slack 4 running under qemu (attachements) with only 6 mb of ram. First image shows it uses about 2 mb + cache/buffers (1 mb). The other one shows the same emulated enivronment running X (1024x768, 24bit color) with twm and one xterm opened. Sorry I forgot to run "free". But it does run, although slow as heck.
