Bounds for generalized Sidon sets

Let Γ be an abelian group and g ≥ h ≥ 2 be integers. A set A ⊂ Γ is a Ch[g]-set if given any set X ⊂ Γ with |X | = h, and any set {k1, . . . , kg } ⊂ Γ , at least one of the translates X + ki is not contained in A. For any g ≥ h ≥ 2, we prove that if A ⊂ {1, 2, . . . , n} is a Ch[g]-set in Z, then |A| ≤ (g − 1)1/hn1−1/h + O(n1/2−1/2h). We show that for any integer n ≥ 1, there is a C3[3]-set A ⊂ {1, 2, . . . , n} with |A| ≥ (4−2/3 + o(1))n2/3. We also show that for any odd prime p, there is a C3[3]-set A ⊂ Fp with |A| ≥ p 2 − p, which is asymptotically best possible. Using the projective norm graphs from extremal graph theory, we show that for each integer h ≥ 3, there is a Ch[h!+1]-set A ⊂ {1, 2, . . . , n}with |A| ≥ (ch + o(1))n1−1/h. A set A is a weak Ch[g]-set if we add the condition that the translates X + k1, . . . , X + kg are all pairwise disjoint. We use the probabilistic method to construct weak Ch[g]-sets in {1, 2, . . . , n} for any g ≥ h ≥ 2. Lastlywe obtain upper bounds on infinite Ch[g]-sequences. We prove that for any infinite Ch[g]-sequence A ⊂ N, we have A(n) = O(n1−1/h(log n)−1/h) for infinitely many n, where A(n) = |A ∩ {1, 2, . . . , n}|. © 2014 Elsevier B.V. All rights reserved.