Square-Difference-Free Sets of Size Omega(n^{0.7334...})
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A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x\ne 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(n^{0.5(1+ \log_{65} 7)}) = Ω(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = Ω(n^{0.5(1+ \log_{205} 12)})= Ω(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.
Fixed important typo: in abstract of paper itself, and on page 3, I had quoted a prior result as being sdf(n) \ge Ω(n^n^{...}) when it should have been sdf(n) \ge Ω(n^{...})
Fixed important typo: in abstract of paper itself, and on page 3, I had quoted a prior result as being sdf(n) \ge Ω(n^n^{...}) when it should have been sdf(n) \ge Ω(n^{...})