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Title: Intersecting families of permutations and other problems in extremal combinatorics
Author: Ellis, David Christopher
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2010
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In this dissertation, we first consider some extremal problems on the symmetric group Sn. A family of permutations A Ì Sn 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 Cameron-Ku conjecture for n sufficiently large. Our proof makes use of the classical representation theory of Sn, or more precisely, (non-Abelian) Fourier Analysis on Sn. In Chapter 2, we consider a natural generalization of the above question. A family of permutations A Ì Sn is said to be t-intersecting 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 t-intersecting family A Ì Sn has size at most (n – t)!. We prove this conjecture using an eigenvalue method, representation theory of Sn, and a combinatorial construction. Ehud Friedgut and Haran Pilpel independently discovered an essentially equivalent proof of the Deza-Frankl 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 k-dimensional subcubes of the n-dimensional 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 k-generator must have size at least k2n/k (1 – o(1)), thereby verifying the above conjecture asymptotically for multiples of k.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral