On the Maximum Number of Integer Colourings with Forbidden Monochromatic Sums
Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{\lceil n/2\rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r \leqslant 5$. Similar results were obtained by Hán and Jiménez in the setting of finite abelian groups.
2021 ◽
Vol 24
(1)
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pp. 263-276
2016 ◽
Vol 58
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pp. 181-202
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2009 ◽
Vol 42
(1)
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pp. 130-136
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