Square-Difference-Free Sets of Size Omega(n^{0.7334...})

Details Square-Difference-Free Sets of Size Omega(n^{0.7334...}) Square-Difference-Free Sets of Size Omega(n^{0.7334...})

2008-04-30

arxiv.org/abs/0804.4892v3

Mathematics

Combinatorics

Fixed important typo: in abstract of paper itself, and on page 3, I had quoted a prior result as being sdf(n) \ge \Omega(n^n^{

Scientific paper

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.

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