The three-cross theorem and the six-cross theorem of Pálfy and Szabó

In 1995, Palfy and Szabo stated the theorem that a projective space satisfies the six-cross theorem if and only if it is Desarguesian. A mistake in their proof is corrected. Moreover, for projective planes, the three-cross theorem of Palfy and Szabo is identified as the Reidemeister condition.