We give simple randomized algorithms leading to new upper bounds for combinatorial problems of Choi and Erdős: For an arbitrary additive group G let $P_n\left(G\right)$ denote the set of all subsets S of G with n elements having the property that 0 is not in S+S. Call a subset A of G admissible with respect to a set S from $P_n\left(G\right)$ if the sum of each pair of distinct elements of A lies outside S. Suppose first that S is a subset of the positive integers in the interval [2n,4n). Denote by f(S) the number of elements in a...