Linear Sets with Five Distinct Differences among Any Four Elements

As a generalization of the concept of Sidon sets, a set of real numbers is called a (4, 5)-set if every four-element subset determines at least five distinct differences. Let g(n) be the largest number such that any n-element (4,5)-set contains a g(n)-element Sidon set (i.e., a subset of g(n) elements with distinct differences). It is shown that (12 + ϵ) n ≤ g(n) ≤ 3n5 + 1, where ϵ is a positive constant. The main result is the lower bound whose proof is based on a Turán-type theorem obtained for sparse 3-uniform hypergraphs associated with (4, 5)-sets.