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On the upper bound of the size of the $$r$$-cover-free families. (English) Zbl 0798.05071
Let $$T(r,n)$$ denote the maximum number of subsets of an $$n$$-set such that no subset is covered by the union of any other $$r$$ subsets (such a system is called $$r$$-cover-free). It is shown that for $$n$$ sufficiently large ${\log_ 2 T(r,n)\over n}\leq 8 {\log_ 2 r\over r^ 2}.$ This comes from a better understanding and proof of a result of A. G. Dyachkov and V. V. Rykov. The central element of this proof is a new set compression algorithm.

##### MSC:
 05D05 Extremal set theory 94B25 Combinatorial codes
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##### References:
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