M ay 2 00 8 Square-Difference-Free Sets of Size Ω ( n 0 . 7334 · · · )

A set A ⊆ N is square-difference free (henceforth SDF) if there do not exist x, y ∈ A, x 6= y, such that |x − y| is a square. Let sdf(n) be the size of the largest SDF subset of {1, . . . , n}. Ruzsa has shown that sdf(n) = Ω(n65 ) = Ω(n0.733077···) We improve on the lower bound by showing sdf(n) = Ω(n205 ) = Ω(n0.7334···) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.