62,396 research outputs found

    A Proof of the Cameron-Ku conjecture

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    A family of permutations A \subset S_n is said to be intersecting if any two permutations in A agree at some point, i.e. for any \sigma, \pi \in A, there is some i such that \sigma(i)=\pi(i). Deza and Frankl showed that for such a family, |A| <= (n-1)!. Cameron and Ku showed that if equality holds then A = {\sigma \in S_{n}: \sigma(i)=j} for some i and j. They conjectured a `stability' version of this result, namely that there exists a constant c < 1 such that if A \subset S_{n} is an intersecting family of size at least c(n-1)!, then there exist i and j such that every permutation in A maps i to j (we call such a family `centred'). They also made the stronger `Hilton-Milner' type conjecture that for n \geq 6, if A \subset S_{n} is a non-centred intersecting family, then A cannot be larger than the family C = {\sigma \in S_{n}: \sigma(1)=1, \sigma(i)=i \textrm{for some} i > 2} \cup {(12)}, which has size (1-1/e+o(1))(n-1)!. We prove the stability conjecture, and also the Hilton-Milner type conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}. One of our key tools will be an extremal result on cross-intersecting families of permutations, namely that for n \geq 4, if A,B \subset S_{n} are cross-intersecting, then |A||B| \leq ((n-1)!)^{2}. This was a conjecture of Leader; it was recently proved for n sufficiently large by Friedgut, Pilpel and the author.Comment: Updated version with an expanded open problems sectio

    Regular Intersecting Families

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    We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families F\mathcal{F} of kk-element subsets of [n]:={1,…,n},[n]:=\{1,\ldots, n\}, such that every element of [n][n] lies in the same (or approximately the same) number of members of F\mathcal{F}. In particular, we show that we can guarantee ∣F∣=o((n−1k−1))|\mathcal{F}| = o({n-1\choose k-1}) if and only if k=o(n)k=o(n).Comment: 15 pages, accepted versio
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