Chen–Chvátal conjecture for arbitrary metric spaces

Establish that every metric space on n points either has a universal line containing the entire space or has at least n distinct lines.

Background

The paper studies the Chen–Chvátal conjecture for metric spaces induced by connected locally connected graphs. Its main theorem proves the conjecture for this graph class, apart from identifying the exceptional graphs with fewer than n lines. The conjecture itself remains unresolved in the general setting of arbitrary finite metric spaces; the paper notes that the strongest general lower bound known is only asymptotic, namely Ω(n^{2/3}) lines when no universal line exists.