Let $f(k)$ be the minimal $N$ such that if $\{1,\dots ,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k)<c^k$ for some constant $c>0$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Your $f(k)$ is exactly the **$k$-th Schur number** [[nomath]](often denoted $S(k)$ in the “minimal $N$” convention)[[/nomath]]: the least $N$ such that every $k$-coloring of ({1,\dots,N}) contains a monochromatic **Schur triple** (a+b=c) [[nomath]](with $a,b$ not necessarily distinct)[[/nomath]]. ([Wikipedia][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.