Erdős Problem #130 — Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle.

Andrásfai and Erdős asked how large the chromatic and clique numbers can be for the integer-distance graph of an infinite planar set in strong general position, especially whether its chromatic number can be infinite.

Known results

• Anning and Erdős showed that an infinite clique cannot occur.
• Kreisel and Kurz found a seven-point finite example in 2008; no larger example is known.
• Structural results show that finite planar integer-distance sets are almost entirely contained in one line or circle; inside $[-N,N]^2$, the exceptional set has size $O((\log N)^{O(1)})$.
• Under strong general position, finite subsets of $[-N,N]^2$ therefore have size $O((\log N)^{O(1)})$.

2024 finite-set structure

Recent structural work gives strong polylogarithmic bounds for finite strong-general-position sets in bounded regions, but it does not determine whether an infinite such set can have finite or infinite chromatic number.

Current status (as of March 2026): the infinite-clique question is settled negatively, but the chromatic-number question and the overall extremal problem remain open.