Uniform intersecting families with large covering number
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A family ℱ has covering number τ if the size of the smallest set intersecting all sets from ℱ is equal to s. Let m(n,k,τ) stand for the size of the largest intersecting family ℱ of k-element subsets of {1,…,n} with covering number τ. It is a classical result of Erdős and Lovász that m(n,k,k)≤ k^k for any n. In this short note, we explore the behaviour of m(n,k,τ) for n<k^2 and large τ. The results are quite surprising: For example, we show that m(k^3/2,k,τ) = (1-o(1))n-1 k-1 for τ≤ k-k^3/4+o(1). At the same time, m(k^3/2,k,τ)<e^-ck^1/2n k if τ>k-1/2k^1/2.
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