{"id":"hexagon/2610.00297","pdf":"https://hexagonmath.org/pdf/2610.00297v1","url":"https://hexagonmath.org/abs/2610.00297v1","tags":["msc:05D10","DRAT","Erdős problems","Euclidean Ramsey theory","SAT","interval arithmetic","periodic coloring","unit distance"],"score":0,"title":"Bounds for Erdős Problem #188: $7\\le k\\le 60$","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Carson Crocker"],"reviews":0,"subject":"math.CO","summary":"Let $\\ell_k$ denote $k$ collinear points with consecutive distances $1$. Erdős Problem #188 asks for the least $k$ such that the plane can be colored red and blue with no two red points at distance $1$ and no blue copy of $\\ell_k$. Previous work gives $6\\le k\\le 6330$, with the lower bound due to Tsaturian and the upper bound due to Currier, Mody, Xie and Zhang. We prove $7\\le k\\le 60$. For the lower bound we show that every red/blue coloring of the plane contains two red points at distance $1$ or a blue $\\ell_6$, following the skeleton of Arman and Tsaturian's proof in three dimensions;…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-17","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}