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

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