On the maximum number of integer colourings with forbidden monochromatic sums

Abstract

Let $f(n,r)$ denote the maximum number of colourings of $A\subseteq \{1,\dots ,n\}$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\{x,y,z\}$ 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 $r\le 5$. Similar results were obtained by Hàn and Jiménez in the setting of finite abelian groups.

24 pages + 3 page appendix