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Séminaire Lotharingien de Combinatoire, B47b (2001), 17 pp.

# Andreas Dress, M. Klucznik, Jack Koolen and Vincent Moulton

#
2kn - Binomial(2k+1,2)

**Abstract.**
It is shown that every cyclic split system **S** defined
on an *n*-set with #**S** > 2*kn* - Binomial(2*k*+1, 2)
for some
*k* <= (*n*-1)/2 always contains a subset of *k*+1 pairwise
incompatible splits provided one has min(*k*,*n* - (2*k*+1))
<= 3.
In addition, some related old and new conjectures are also discussed.

Received: March 25, 2001; Revised: August 8, 2001;
Accepted: November 12, 2001.

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