On Sequences of Positive Integers Containing No p Terms in Arithmetic Progression

A famous conjecture of Erd¨os asserts that if A is a subset of the positive integers having the property that ∑a∈A 1/a = ∞, then A must contain arithmetic progressions of arbitrarily large length. We use topological ideas to show that if this conjecture is true then there must exist a subset M_p of positive integers with no p terms in arithmetic progression with the property that among all such subsets, M_p maximizes the sum of the reciprocals of its elements.