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On the maximum number of integer colourings with forbidden monochromatic sums
- Publication Year :
- 2017
-
Abstract
- 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 $r \leq 5$. Similar results were obtained by H\`an and Jim\'enez in the setting of finite abelian groups.<br />Comment: 24 pages + 3 page appendix
- Subjects :
- Class (set theory)
Logarithm
Discrete Mathematics
Applied Mathematics
Diskret matematik
Theoretical Computer Science
Combinatorics
11B75, 05D99
Computational Theory and Mathematics
Integer
Annan matematik
FOS: Mathematics
Discrete Mathematics and Combinatorics
Mathematics - Combinatorics
Geometry and Topology
Monochromatic color
Combinatorics (math.CO)
Abelian group
Other Mathematics
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....763d68a26a69c5f373a9a803a8ab0f4e