# A new generalization of the Erdős-Ko-Rado theorem

## Abstract

Let ℱ be a family ofk-subsets of ann-set. Lets be a fixed integer satisfyingks≦3k. Suppose that forF 1,F 2,F 3 ∈ ℱ |F 1F 2F 3|≦s impliesF 1F 2F 3 ≠ 0. Katona asked what is the maximum cardinality,f(n, k, s) of such a system. The Erdős-Ko-Rado theorem impliesf(n, k, s)=$$\left( {_{k - 1}^{n - 1} } \right)$$ fors=3k andn≧2k. In this paper we show thatf(n, k, s)=$$\left( {_{k - 1}^{n - 1} } \right)$$ holds forn>n 0(k) if and only ifs≧2k.

Equality holds only if every member of ℱ contains a fixed element of the underlying set.

Further we solve the problem fork=3,s=5,n≧3000. This result sharpens a theorem of Bollobás.

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Dedicated to Paul Erdős on his seventieth birthday

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