4n- 10.

We show that the maximal numberK2(n) of splits in a 2-compatiblesplit system on ann-set is exactly 4n- 10, for everyn>3.SinceK2(n) =CF3(n)/2 whereCF3(n) is the maximal number of members in any 3-cross-freecollection of (proper) subsets of ann-set, this gives a definitive answer to a question raised in 1979 by A. Karzanov who asked whetherCF3(n) is, as a function ofn, of typeO(n).Karzanov’s question was answered first by P. Pevzner in 1987 who showedK2(n) = 6n, a result that was improved by T. Fleiner in 1998 who showedK2(n) = 5n.The argument given in the paper below establishes that the even slightly stronger inequalityK2(n) = 4n- 10 holds for everyn>3; the identityK2(n) = 4n- 10 (n>3) then follows in conjunction with a result from a previous paper that impliesK2(n) = 4n- 10. In that paper, it was also mentioned that-in analogy to well known results regarding maximalweakly compatiblesplit systems-2-compatible split systems of maximal cardinality 4n- 10 should be expected to becyclic. Luckily, our approach here permits us also to corroborate this expectation. As a consequence, it is now possible to generate all 2-compatible split systems on ann-set (n>3) that have maximal cardinality 4n- 10 (or, equivalently, all 3-cross-free set systems that have maximal cardinality 8n- 20) in a straight forward, systematic manner. [ABSTRACT FROM AUTHOR]