Square-Difference-Free Sets of Size Omega(n^{0.7334...})
Combinatorics
2008-05-08 v3
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.
Keywords
Cite
@article{arxiv.0804.4892,
title = {Square-Difference-Free Sets of Size Omega(n^{0.7334...})},
author = {Richard Beigel and William Gasarch},
journal= {arXiv preprint arXiv:0804.4892},
year = {2008}
}
Comments
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^{...})