Title:

Intersecting families of permutations and other problems in extremal combinatorics

In this dissertation, we first consider some extremal problems on the symmetric group S_{n}. A family of permutations A Ì S_{n} is said to be intersecting if any two permutations in A agree in at least one point, i.e. for any s, p Î A , there is some i Î [n] such that s(i) = p(i). In Chapter 1, we prove the CameronKu conjecture for n sufficiently large. Our proof makes use of the classical representation theory of S_{n}, or more precisely, (nonAbelian) Fourier Analysis on S_{n}. In Chapter 2, we consider a natural generalization of the above question. A family of permutations A Ì S_{n} is said to be tintersecting if any two permutations in A agree in at least t points, i.e. for any s, p Î A, {i Î [n] : s(i) = p(i)}³t. Deza and Frankl conjectured in 1977 that if n is sufficiently large depending on t, a tintersecting family A Ì S_{n} has size at most (n – t)!. We prove this conjecture using an eigenvalue method, representation theory of S_{n}, and a combinatorial construction. Ehud Friedgut and Haran Pilpel independently discovered an essentially equivalent proof of the DezaFrankl conjecture, and we have now written a joint paper. In Chapter 3, we consider the problem of finding the maximum possible size of a family of kdimensional subcubes of the ndimensional cube {0,1}^{n}, none of which is contained in the union of the others (we call such a family irredundant). In Chapter 4, we prove a generalization of a theorem of Alon and Frankl in order to show that for fixed k, a kgenerator must have size at least k2^{n/k} (1 – o(1)), thereby verifying the above conjecture asymptotically for multiples of k.
