Some combinatorial theorems with an application to a problem in number theory

The main object of this thesis is to study the following extremal problem in number theory: Let n and k be integers satisfying n ≥ k ≥ 3. Denote by f(n,k) the largest positive integer for which there exists a set S of f(n,k) integers satisfying -- (i) S ⊑ { 1,2...,n } and -- (ii) no k numbers in S have pairwise the same greatest common divisor. -- We investigate the behaviour of f(n,k) in the case where k → ∞ with n. In particular we obtain estimates for f(n, [logαn]) for fixed α > 0 and f(n,[nα]) for fixed α, 0 < α < 1. -- In the course of our investigations we make use of certain intersection theorems for systems of finite sets. We also include a number of new results concerning these theorems.