Dense Admissible Sets

Call a set of integers {b1, b2, . . . , bk} admissible if for any prime p, at least one congruence class modulo p does not contain any of the bi. Let ρ ∗(x) be the size of the largest admissible set in [1, x]. The Prime k-tuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b1, n+b2, . . . n+bk are simultaneously prime. In 1974, Hensley and Richards [3] showed that ρ∗(x) > π(x) for x sufficiently large, which shows that the Prime k-tuples Conjecture is inconsistent with a conjecture of Hardy and Littlewood that for all integers x, y ≥ 2, π(x+ y) ≤ π(x) + π(y). In this paper we examine the behavior of ρ∗(x), in particular, the point at which ρ∗(x) first exceeds π(x), and its asymptotic growth.