Square-Difference-Free Sets of Size Omega(n^{0.7334...})
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2008
- DOI:
- 10.48550/arXiv.0804.4892
- arXiv:
- arXiv:0804.4892
- Bibcode:
- 2008arXiv0804.4892B
- Keywords:
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- Mathematics - Combinatorics;
- 05D10
- E-Print:
- 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^{...}) when it should have been sdf(n) \ge \Omega(n^{...})