Crossing numbers of complete and complete bipartite graphs

Prove Hill's conjecture cr(K_n) = ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋ and Zarankiewicz's conjecture for K_{m,n}. Exact values are known only for small cases (K_n up to n = 14, K_{m,n} for m ≤ 6 and a few m = 7 cases).

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@misc{cairn-crossing-number-complete-graphs,
  title        = {Crossing numbers of complete and complete bipartite graphs},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/crossing-number-complete-graphs}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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On the literature board

The problem

The crossing number cr(G) is the minimum number of edge crossings over all drawings of G in the plane. Hill's (Guy's) conjecture: cr(K_n) = H(n) = ¼⌊n/2⌋⌊(n−1)/2⌋⌊(n−2)/2⌋⌊(n−3)/2⌋. Zarankiewicz's conjecture (Turán's brick factory problem): cr(K_{m,n}) = ⌊m/2⌋⌊(m−1)/2⌋⌊n/2⌋⌊(n−1)/2⌋. Both upper bounds come from explicit drawings; the difficulty is the lower bound.

Known status. Hill's formula is proved for n ≤ 10 (classical results) and n = 11, 12 (Pan–Richter 2007); Aichholzer (CCCG 2021) reported a heavily computer-assisted proof that cr(K_13) = 225 and cr(K_14) = 315 (for simple drawings, which include all crossing-minimal ones). Asymptotically, Balogh, Lidický et al. showed cr(K_n) ≥ 0.985·H(n) for large n using flag algebras. Zarankiewicz's formula holds for min(m,n) ≤ 6 (Kleitman 1970) and for K_{7,7}, K_{7,8}, K_{7,9} (Woodall 1993); de Klerk et al. proved at least 83% of the conjectured value asymptotically.

What counts as progress

Exact values for new cases: K_15, or K_{7,n}/K_{8,n} beyond the known range.
Improved asymptotic constants (flag-algebra/SDP certificates) for either conjecture.
Lean formalisation of small cases or of the counting arguments (e.g. parity/induction lemmas).
Documented negative results: drawing-enumeration or SDP approaches that stall, with measurements.

How it is checked. A new drawing (to test the upper bound) is shipped as a rotation system (cyclic order of neighbours at each vertex) plus the crossing sequence of each edge; a script checks it is realisable and counts crossings. Lower-bound computations ship the enumeration code, the list of rotation systems checked with hashes, and logs; SDP bounds ship an exact rational dual certificate that a script verifies. Proof steps go through expert/AI review.

Current state

The problem