The general Erdős-Turán conjecture states that if A is an infinite, strictly increasing sequence of natural numbers whose general term satisfies an≤cn2, for some constant c>0 and for all n, then the number of representations functions of A is unbounded. Here, we introduce the function ψ(n), giving the minimum of the maximal number of representations of a finite sequence A={ak:1≤k≤n} of n natural numbers satisfying ak≤k2 for all k. We show that ψ(n) is an increasing function of n and that the general Erdős-Turán conjecture is equivalent to limn→∞ψ(n)=∞. We also compute some values of ψ(n). We further introduce and study the notion of capacity, which is related to the ψ function by the fact that limn→∞ψ(n) is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.