Generalized additive bases, Königʹs lemma, and the Erd s–Turán conjecture

Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h, let rA(n,h) denote the number of representations of n in the form n=a1+a2+ +ah, where a1,a2,…,ah A and a1 a2 ah. The infinite set A is called a basis of order h if rA(n,h) 1 for every nonnegative integer n. Erd s and Turán conjectured that lim supn→∞ rA(n,2)=∞ for every basis A of order 2. This paper introduces a new class of additive bases and a general additive problem, a special case of which is the Erd s–Turán conjecture. Königʹs lemma on the existence of infinite paths in certain graphs is used to prove that this general problem is equivalent to a related problem about finite sets of nonnegative integers.