P\'osa's Conjecture for graphs of order at least 2\times 10^8

In 1962 P'osa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of P'osa’s Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than 2n/3. Still in 1996, Koml'os, S'ark"ozy, and Szemer'edi proved P'osa’s Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n > n_0, where n_0 is a very large constant. Here we show without using these lemmas that n_0=2\times 10^8 is sufficient. We are motivated by the recent work of Levitt, Szemer'edi and S'ark"ozy, but our methods are based on techniques that were available in the 90’s.