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• W2498056876 abstract "Given a linear equation $mathcal{L}$, a set $A subseteq [n]$ is $mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $mathcal{L}$. In this paper we consider the following three general questions: (i) What is the size of the largest $mathcal{L}$-free subset of $[n]$? (ii) How many $mathcal{L}$-free subsets of $[n]$ are there? (iii) How many maximal $mathcal{L}$-free subsets of $[n]$ are there? We completely resolve (i) in the case when $mathcal{L}$ is the equation $px+qy=z$ for fixed $p,qin mathbb N$ where $pgeq 2$. Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations $mathcal{L}$, thereby refining a special case of a result of Green. We also give various bounds on the number of maximal $mathcal{L}$-free subsets of $[n]$ for three-variable homogeneous linear equations $mathcal{L}$. For this, we make use of container and removal lemmas of Green." @default.
• W2498056876 created "2016-08-23" @default.
• W2498056876 creator A5038904202 @default.
• W2498056876 creator A5086607043 @default.
• W2498056876 date "2016-07-28" @default.
• W2498056876 modified "2023-09-27" @default.
• W2498056876 title "On solution-free sets of integers" @default.
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• W2498056876 hasPublicationYear "2016" @default.
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