Journal or Book Title
Borrowing Laszlo Szekely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr(Kn) of the complete graph Kn is H(n):=14⌊n2⌋⌊n−12⌋⌊n−22⌋⌊n−32⌋, for all n≥3. This has been verified only for n≤12. Using flag algebras, Norin and Zwols obtained the best known asymptotic lower bound for the crossing number of complete bipartite graphs, from which it follows that for every sufficiently large n, cr(Kn)>0.905H(n). Also using flag algebras, we prove that asymptotically cr(Kn) is at least 0.985H(n). We also show that the spherical geodesic crossing number of Kn is asymptotically at least 0.996H(n).
Balogh, Josef; Lidicky, Bernard; and Salazar, Gelasio, "Closing in on Hill's conjecture" (2017). Mathematics Publications. 182.