Dense edge-magic graphs and thin additive bases

A graph G of order n and size m is edge-magic if there is a bijection l : V (G) ∪E(G) → [n+m] such that all sums l(a) + l(b) + l(ab), ab ∈ E(G), are the same. We present new lower and upper bounds on M(n), the maximum size of an edge-magic graph of order n, being the first to show an upper bound of the form M(n) ≤ (1 − ǫ) ( n 2 ) . Concrete estimates for ǫ can be obtained by knowing s(k, n), the maximum number of distinct pairwise sums that a k-subset of [n] can have. So, we also study s(k, n), motivated by the above connections to edge-magic graphs and by the fact that a few known functions from additive number theory can be expressed via s(k, n). For example, our estimate s(k, n) ≤ n+ k2 ( 1 4 − 1 (π + 2) + o(1) ) implies new bounds on the maximum size of quasi-Sidon sets, a problem posed by Erdős and Freud [J. Number Th. 38 (1991) 196–205]. The related problem for differences is considered as well.