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Claim. For a real number $u$ write $\{u\}$ for its fractional part, read in $\mathbb R/\mathbb Z$. An arc is a nonempty connected subset $I$ of $\mathbb R/\mathbb Z$, open, closed or half-open, and it is sum-free if $t+t'\notin I$ for all $t,t'\in I$. Let $N$ and $k$ be positive integers and $\{1,\ldots,N\}=A_1\cup\cdots\cup A_k$, for instance the color classes of a $k$-coloring. Suppose that for each $i$ there are a real number $\theta_i$ and a sum-free arc $I_i$ such that $\{\theta_ix\}\in I_i$ for every $x\in A_i$. Then

$$N+1\le8^k.$$

Each set $\{x:\{\theta x\}\in I\}$ with $I$ a sum-free arc is sum-free, so a coloring of this kind has no monochromatic solution to $a+b=c$, equal summands allowed. Thus the least $N$ such that $\{1,\ldots,N\}$ has no $k$-coloring of this kind is at most $8^k$.

Covers. The exponential question of Problem 483 for the $k$-colorings of $\{1,\ldots,N\}$ in which each color class lies in a single sum-free Bohr arc $\{x:\{\theta_ix\}\in I_i\}$: for these colorings the exponential bound holds, since none exists once $N\ge8^k$. The class contains Schur's tripling coloring by $k$ colors, of $\{1,\ldots,(3^k-1)/2\}$. A $k$-coloring of $\{1,\ldots,N\}$ with $N\ge8^k$ and no monochromatic solution therefore has a class lying in no single sum-free Bohr arc. The bound does not apply to general colorings: $f(k)$ itself is not bounded by it, and the question whether $f(k)<c^k$, with the estimate of $f(k)$ and whether $f(k)^{1/k}$ tends to a finite limit, remains open.

Standing. L262 is author-recorded at tier 0: an elementary complete argument, a Dirichlet pigeonhole bound with an integer dilation moving each sum-free arc away from zero, using no external theorem, without independent examination or a Lean proof. The constant $8$ is not optimized, and no literature search for the observation is recorded. Catalog acceptance is pending.

Depends on. L262 (tier 0), which consumes no other native claim and no external theorem.