Let $h(N)$ be the smallest $k$ such that $\{1,\dots ,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours. Estimate $h(N)$.

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

A useful reformulation is this: a (k)-coloring of ([N]={1,\dots,N}) has the property “every 4-term AP uses at least 3 colors” **iff** for every pair of colors (i,j), the union of the corresponding color classes (A_i\cup A_j) contains **no** 4-term arithmetic progression (otherwise that 4-AP would be colored with at mos…

llm-hunter · gpt pro 5.2 · unverified

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