For every integer n≥ 3 and every set of n distinct lines in the real projective plane RP^2 that are not all concurrent, there exist at least three points of RP^2 incident with exactly two of the lines.

Melchior’s ordinary-points problem

The problem seeks a simpler geometric proof of Melchior’s conclusion that every nontrivial finite arrangement of real projective lines has at least three ordinary points. Melchior proved the underlying result in 1940, though one source dates it 1941; it predates Erdős’s 1943 restatement.

Known results

Melchior, 1940 or 1941: Euler’s polyhedral formula yields at least three ordinary points.
Erdős, 1943: restated the ordinary-line problem.
Grünwald, 1944: published a solution of the ordinary-line problem.

August 25, 2026 visual proof

The paper In Search of Melchior’s Ordinary Points reports a visual proof of the three-ordinary-points conclusion and presents it as a simpler geometric route to the stronger Sylvester–Gallai phenomenon. The underlying theorem was already known, so this is progress on the requested proof method rather than a new resolution; the proof has not been independently verified here.

Current status (as of August 2026): Melchior’s three-ordinary-points theorem is settled, while the newly reported visual proof is claimed but unverified.

Melchior’s ordinary-points problem

Sylvester's ordinary-line theorem

Primary source

Additional references

Progress summary

Solutions 0