Crossing numbers of complete and complete bipartite graphs

Zarankiewicz's formula for cr(K_{m,n}) and the Harary-Hill formula for cr(K_n), proved for unrestricted drawings: normal drawings, the cycle pairing of signed crossings, interpolation, a subspace intersection inequality, and the matching drawings (OpenAI, 2026).

Eight lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Drawings and crossing numbers
2. Polygonal drawings
3. Signed crossings and cycles
4. Interpolation
5. An intersection inequality for subspaces
6. The Zarankiewicz formula
7. The lower bound for complete graphs
8. Two-page drawings and the Harary–Hill formula

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Compactness, uniform continuity, the Lebesgue covering lemma and the inverse function theorem (core courses Real Analysis I and II).
• Rank, determinants and bilinear forms (core course Linear Algebra).
• Polynomials in several variables (core course Abstract Algebra I).