Determine g(n): the minimum over configurations of n points in the plane of the number of distinct pairwise distances they determine. Erdős conjectured g(n) ≈ n/√log n (grid constructions achieve this); the question is whether g(n) ≥ c·n for absolute c.
One of Erdős's signature questions, mother to half of discrete geometry: Moser (1952), Chung and a generation of lattice methods crawled upward until Larry Guth and Nets Katz (2010) introduced polynomial partitioning and proved g(n) ≥ cn/log n — landing within a single logarithm of the grid bound and winning massive recognition for algebraic geometry methods in combinatorics. That logarithm has not moved since; closing it (or finding super-grid configurations) remains open, and the machinery invented along the way now powers the rest of this directory's geometry shelf.
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