Geodesic
On the maximum number of integer colourings with forbidden monochromatic sums
Hong Liu1 ; Maryam Sharifzadeh ; Katherine Staden2
1University of Warwick
2University of Oxford
The electronic journal of combinatorics, Tome 28 (2021) no. 1
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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.
DOI : 10.37236/8824
Classification : 11B75, 11B13
Mots-clés : maximum number of integer colorings, forbidden monochromatic sums

Hong Liu  1   ; Maryam Sharifzadeh    ; Katherine Staden  2

1 University of Warwick
2 University of Oxford 
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Hong Liu; Maryam Sharifzadeh; Katherine Staden. On the maximum number of integer colourings with forbidden monochromatic sums. The electronic journal of combinatorics, Tome 28 (2021) no. 1. doi: 10.37236/8824

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