A Proof of the Cameron-Ku conjecture

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

The chromatic number of the q-Kneser graph for large q

We obtain a new weak Hilton-Milner type result for intersecting families ofk-spaces inF2kq, which improves several known results. In particular the chromaticnumber of theq-Kneser graphqKn:kwas previously known forn >2k(except forn= 2k+1 andq= 2) ork 5, so that the only remaining open cases are (n, k) = (2k, k) withq∈{2,3,4}and (n, k) = (2k+ 1, k) withq= 2