I translated this as "I don't know either František or Matěj" and, fortunately, that was accepted.
But could someone explain, please, why there is only one "ani" in this sentence, if the main translation uses "neither/nor"? I thought that would be expressed by "ani/ani." Thanks!
Considering only Czech, the first "ani" is just stressing it, but does not change the meaning in any way. Whether there should be any difference in the translations to English ... I don't know.
Hmm. The other English alternative I've though of would be "I don't know František OR Matěj." But would that use nebo instead of ani on the Czech side? If it would, then I think we have only the "(know) neither/nor" and "(don't know) either/or" options.