%0 Journal Article
%T Square-Difference-Free Sets of Size Omega(n^{0.7334...})
%A Richard Beigel
%A William Gasarch
%J Mathematics
%D 2008
%I arXiv
%X 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) = \Omega(n^{0.5(1+ \log_{65} 7)}) = \Omega(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = \Omega(n^{0.5(1+ \log_{205} 12)})= \Omega(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.
%U http://arxiv.org/abs/0804.4892v3