Bounds for Erdős Problem #188: $7\le k\le 60$
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;…
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