Geodesic
An improved lower bound related to the Furstenberg-Sárközy theorem
The electronic journal of combinatorics, Tome 22 (2015) no. 1
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Let $D(n)$ denote the cardinality of the largest subset of the set $\{1,2,\ldots,n\}$ such that the difference of no pair of distinct elements is a square. A well-known theorem of Furstenberg and Sárközy states that $D(n)=o(n)$. In the other direction, Ruzsa has proven that $D(n) \gtrsim n^{\gamma}$ for $\gamma = \frac{1}{2}\left( 1 + \frac{\log 7}{\log 65} \right) \approx 0.733077$. We improve this to $\gamma = \frac{1}{2}\left( 1 + \frac{\log 12}{\log 205} \right) \approx 0.733412$.
DOI : 10.37236/4656
Classification : 05D10, 05B10
Mots-clés : Ramsey theory, squares, difference set

Mark Lewko  1

1 UCLA 
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     author = {Mark Lewko},
     title = {An improved lower bound related to the {Furstenberg-S\'ark\"ozy} theorem},
     journal = {The electronic journal of combinatorics},
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Mark Lewko. An improved lower bound related to the Furstenberg-Sárközy theorem. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4656

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