On the maximum number of translates

Given a finite set P ⊆ Rd, called a pattern, tP (n) denotes the maximum number of translated copies of P determined by n points in Rd. We give the exact value of tP (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that tP (n) = n −mr (n), where r is the rational affine dimension of P , and mr (n) is the r-Kruskal-Macaulay function. We note that almost all patterns in Rd are rational simplices. The function tP (n) is also determined exactly when |P | ≤ 3 or when P has rational affine dimension one and n is large enough. We establish the equivalence finding tP (n) and the maximum number sR (n) of scaled copies of a suitable pattern R ⊆ R+ determined by n positive reals. As a consequence, we show that sAk (n) = n − Θ ( n1−1/π(k) ) , where Ak is an arithmetic progression of size k, and π (k) is the number of primes less than or equal to k. AMS Subject Classification: 52C10 (Erdős Problems), 05C35 (Extremal Problems), 05B35 (Geometric Lattices)