A Treatise on the Binomial Theorem

OF THE DISSERTATION A treatise on the binomial theorem by PATRICK DEVLIN Dissertation Director: Jeff Kahn This dissertation discusses four problems taken from various areas of combinatorics— stability results, extremal set systems, information theory, and hypergraph matchings. Though diverse in content, the unifying theme throughout is that each proof relies on the machinery of probabilistic combinatorics. The first chapter offers a summary. In the second chapter, we prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose A is an n× n matrix over C (resp. R), and let P denote the set of n× n matrices over C (resp. R) that can be written as a permutation matrix times a unitary diagonal matrix. Then it is known that the permanent of A satisfies |perm(A)| ≤ ‖A‖2 with equality iff A/‖A‖2 ∈ P (where ‖A‖2 is the operator 2-norm of A). We show a stability version of this result asserting that unless A is very close (in a particular sense) to one of these extremal matrices, its permanent is exponentially smaller (as a function of n) than ‖A‖2 . In particular, for any fixed α, β > 0, we show that |perm(A)| is exponentially smaller than ‖A‖2 unless all but at most αn rows contain entries of modulus at least ‖A‖2(1− β). In the third chapter, we prove a randomized result extending the classical Erdős– Ko–Rado theorem. Namely, let Kp(n, k) denote the random subgraph of the usual