Partitioning all k-subsets into r-wise intersecting families
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Let r ≥ 2, n and k be integers satisfying k ≤r-1/rn. In the original arXiv version of this note we suggested a conjecture that the family of all k-subsets of an n-set cannot be partitioned into fewer than ⌈ n-r/r-1(k-1) ⌉ r-wise intersecting families. We noted that if true this is tight for all values of the parameters, that the case r=2 is Kneser's conjecture, proved by Lovász, and observed that the assertion also holds provided r is either a prime number or a power of 2. We have recently learned, however, that the assertion of the conjecture for all values of the parameters follows from a recent result of Azarpendar and Jafari <cit.>.
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