volume 31, issue 2, P251-255 2004
DOI: 10.1007/s00454-003-2859-z
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Abstract: We prove that every set system of bounded VC-dimension has a fractional Helly property. More precisely, if the dual shatter function of a set system F is bounded by o(m k ), then F has fractional Helly number k. This means that for every α > 0 there exists a β > 0 such that if F 1 , F 2 , . . . , F n ∈ F are sets with i∈I F i = ∅ for at least α n k sets I ⊆ {1, 2, . . . , n} of size k, then there exists a point common to at least βn of the F i . This further implies a ( p, k)-theorem: for every F as above and…

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