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- W2765793708 abstract "An $r$-coloring of a subset $A$ of a finite abelian group $G$ is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements $a,b,cin A$ with $a+b=c$. We investigate $kappa_{r,G}$, the maximum number of sum-free $r$-colorings admitted by subsets of $G$, and our results show a close relationship between $kappa_{r,G}$ and largest sum-free sets of $G$. Given a sufficiently large abelian group $G$ of type $I$, i.e., $|G|$ has a prime divisor $q$ with $qequiv 2pmod 3$. For $r=2,3$ we show that a subset $Asubset G$ achieves $kappa_{r,G}$ if and only if $A$ is a largest sum-free set of $G$. For even order $G$ the result extends to $r=4,5$, where the phenomenon persists only if $G$ has a unique largest sum-free set. On the contrary, if the largest sum-free set in $G$ is not unique then $A$ attains $kappa_{r,G}$ if and only if it is the union of two largest sum-free sets (in case $r=4$) and the union of three (independent) largest sum-free sets (in case $r=5$). Our approach relies on the so called container method and can be extended to larger $r$ in case $G$ is of even order and contains sufficiently many largest sum-free sets." @default.
- W2765793708 created "2017-11-10" @default.
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- W2765793708 date "2017-10-23" @default.
- W2765793708 modified "2023-10-18" @default.
- W2765793708 title "Maximum number of sum-free colorings in finite abelian groups" @default.
- W2765793708 hasPublicationYear "2017" @default.
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